Numerical approximation and simulation of the stochastic wave equation on the sphere

Abstract

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrödinger equation on the unit sphere.

1 Introduction

The recent years have witnessed a strong interest in the theoretical study of (regularity) properties and the simulation of random fields, especially the ones that are defined by stochastic partial differential equations (SPDEs) on Euclidean spaces. This increase in the interest in random fields is due to the huge demand from applications as diverse as models for the motion of a strand of DNA floating in a fluid [\(Y_{\ell , m}:[0,\pi ] \times [0,2\pi ) \rightarrow {\mathbb {C}},\) which are given by

$$\begin{aligned} Y_{\ell , m}(\vartheta , \varphi ) = \sqrt{\frac{2\ell + 1}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}} P_{\ell , m}(\cos \vartheta ) \ \mathrm {e}^{\mathrm {i}m\varphi } \end{aligned}$$

for $\ell \in {\mathbb {N}}_0,$ $m = 0,\ldots , \ell ,$ and $(\vartheta ,\varphi ) \in [0,\pi ] \times [0,2\pi )$ and by

$$\begin{aligned} Y_{\ell , m} = (-1)^m \overline{Y_{\ell , -m}} \end{aligned}$$

for $\ell \in {\mathbb {N}}$ and $m=-\ell , \ldots ,-1.$ It is well-known that the spherical harmonics form an orthonormal basis of $L^2({\mathbb {S}}^2),$ the subspace of real-valued functions in $L^2({\mathbb {S}}^2;{\mathbb {C}}).$ In what follows we set for $y \in {\mathbb {S}}^2$

$$\begin{aligned} Y_{\ell , m}(y) = Y_{\ell , m}(\vartheta ,\varphi ), \end{aligned}$$

where $y = (\sin \vartheta \cos \varphi , \sin \vartheta \sin \varphi , \cos \vartheta ),$ i.e., we identify (with a slight abuse of notation) Cartesian and angular coordinates of the point $y\in {\mathbb {S}}^2.$ Furthermore we denote by $\sigma$ the Lebesgue measure on the sphere which admits the representation

$$\begin{aligned} d\sigma (y) = \sin \vartheta \, d\vartheta \, d\varphi \end{aligned}$$

for $y = (\sin \vartheta \cos \varphi , \sin \vartheta \sin \varphi , \cos \vartheta )\in {\mathbb {S}}^2.$

The spherical Laplacian, also called Laplace–Beltrami operator, is given in terms of spherical coordinates similarly to Section 3.4.3 in [34] by

$$\begin{aligned} \varDelta _{{\mathbb {S}}^2} = (\sin \vartheta )^{-1} \frac{\partial }{\partial \vartheta } \left( \sin \vartheta \, \frac{\partial }{\partial \vartheta } \right) + (\sin \vartheta )^{-2} \frac{\partial ^2}{\partial \varphi ^2}. \end{aligned}$$

It is well-known (see, e.g., Theorem 2.13 in [35]) that the spherical harmonic functions ${\mathscr {Y}}$ are the eigenfunctions of $\varDelta _{{\mathbb {S}}^2}$ with eigenvalues $(-\ell (\ell +1), \ell \in {\mathbb {N}}_0),$ i.e.,

$$\begin{aligned} \varDelta _{{\mathbb {S}}^2} Y_{\ell , m} = - \ell (\ell +1) Y_{\ell , m} \end{aligned}$$

for all $\ell \in {\mathbb {N}}_0,$ $m = -\ell , \ldots ,\ell .$

To characterize the regularity of solutions to SPDEs in what follows, we introduce the Sobolev space on ${\mathbb {S}}^2$ for a smoothness index $s\in {\mathbb {R}}$

$$\begin{aligned} H^s({\mathbb {S}}^2)=(\text{Id}-\varDelta _{{\mathbb {S}}^2})^{-s/2}L^2({\mathbb {S}}^2) \end{aligned}$$

together with its norm

$$\begin{aligned} \Vert f\Vert _{H^s({\mathbb {S}}^2)}=\Vert (\text{Id}-\varDelta _{{\mathbb {S}}^2})^{s/2}f\Vert _{L^2({\mathbb {S}}^2)} \end{aligned}$$

for some $f\in H^s({\mathbb {S}}^2)$ with $H^0({\mathbb {S}}^2) = L^2({\mathbb {S}}^2).$ Further definitions and information on these spaces can be found for instance in [36].

Furthermore, we work on $L^p(\varOmega ;H^s({\mathbb {S}}^2))$ with norm

$$\begin{aligned} \Vert Z\Vert _{L^p(\varOmega ;H^s({\mathbb {S}}^2))} = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert Z\Vert _{H^s({\mathbb {S}}^2)}^p\right] ^{1/p} \end{aligned}$$

for finite $p \ge 1$ and are now in place to introduce the following definitions:

A ${\mathscr {A}}\otimes {\mathscr {B}}({\mathbb {S}}^2)$-measurable map** $Z:\varOmega \times {\mathbb {S}}^2 \rightarrow {\mathbb {R}}$ is called a real-valued random field on the unit sphere. Such a random field is called Gaussian if for all $k \in {\mathbb {N}}$ and $x_1, \ldots , x_k \in {\mathbb {S}}^2,$ the multivariate random variable $(Z(x_1),\ldots ,Z(x_k))$ is multivariate Gaussian distributed. Finally, such a random field is called isotropic if its covariance function only depends on the distance d(x, y),  for $x,y\in {\mathbb {S}}^2.$

We recall Theorem 2.3 and Lemma 5.1 in [32] on the series expansions of isotropic Gaussian random fields on the sphere.

Lemma 1

A centered, isotropic Gaussian random field Z has a converging Karhunen–Loève expansion

$$\begin{aligned} Z = \sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell a_{\ell , m} Y_{\ell , m} \end{aligned}$$

(1)

with $a_{\ell , m} = (Z,Y_{\ell , m})_{L^2({\mathbb {S}}^2)}$ and $A_\ell = {{\,\mathrm{{\mathbb {E}}}\,}}[a_{\ell , m} \overline{a_{\ell , m}}]$ for all $m=-\ell ,\ldots ,\ell ,$ where $(A_\ell ,\ell \in {\mathbb {N}}_0)$ is called the angular power spectrum of Z. For $\ell \in {\mathbb {N}},$ $m = 1,\ldots ,\ell ,$ and $\vartheta \in [0,\pi ]$ set

$$\begin{aligned} L_{\ell , m}(\vartheta ) = \sqrt{\frac{2\ell +1}{4\pi } \frac{(\ell -m)!}{(\ell +m)!}} P_{\ell , m}(\cos \vartheta ). \end{aligned}$$

The expansion (1) converges in $L^2(\varOmega \times {\mathbb {S}}^2)$ and in $L^2(\varOmega )$ for all $y\in {\mathbb {S}}^2.$

Then for $y= (\sin \vartheta \cos \varphi , \sin \vartheta \sin \varphi , \cos \vartheta ),$ it holds

$$\begin{aligned} Z(y) = \sum _{\ell =0}^\infty \left( \sqrt{A_\ell } X^1_{\ell , 0} L_{\ell , 0}(\vartheta ) + \sqrt{2A_\ell } \sum _{m=1}^\ell L_{\ell , m}(\vartheta ) (X^1_{\ell , m} \cos (m\varphi ) + X^2_{\ell , m} \sin (m \varphi ))\right) \end{aligned}$$

in law, where $(X^i_{\ell , m}, i=1,2, \ell \in {\mathbb {N}}_0, m=0,\ldots , \ell )$ is a sequence of independent, real-valued, standard normally distributed random variables and $X^2_{\ell , 0} = 0$ for $\ell \in {\mathbb {N}}_0.$

In order to simulate solutions to the stochastic wave equation on the sphere, we need to approximate the driving noise which can be generated by a sequence of Gaussian random fields. We choose to truncate the above series expansion for an index $\kappa \in {\mathbb {N}}$ and set

$$\begin{aligned} Z^\kappa (y) = \sum _{\ell =0}^\kappa \left( \sqrt{A_\ell } X^1_{\ell , 0} L_{\ell , 0}(\vartheta ) + \sqrt{2A_\ell } \sum _{m=1}^\ell L_{\ell , m}(\vartheta ) (X^1_{\ell , m} \cos (m\varphi ) + X^2_{\ell , m} \sin (m \varphi ))\right) , \end{aligned}$$

where we recall $y= (\sin \vartheta \cos \varphi , \sin \vartheta \sin \varphi , \cos \vartheta )$ and $(\vartheta ,\varphi ) \in [0,\pi ]\times [0,2\pi ).$

The above lemma then allows us to present the following results on $L^p(\varOmega ;L^2({\mathbb {S}}^2))$ convergence and ${\mathbb {P}}$-almost sure convergence of the truncated series which are proven in Theorem 5.3 and Corollary 5.4 in [32].

Theorem 1

Let the angular power spectrum $(A_\ell , \ell \in {\mathbb {N}}_0)$ of the centered, isotropic Gaussian random field Z decay algebraically with order $\alpha >2,$ i.e., there exist constants $C>0$ and $\ell _0 \in {\mathbb {N}}$ such that $A_\ell \le C \cdot \ell ^{-\alpha }$ for all $\ell > \ell _0.$ Then the series of approximate random fields $(Z^\kappa , \kappa \in {\mathbb {N}})$ converges to the random field Z in $L^p(\varOmega ;L^2({\mathbb {S}}^2))$ for any finite $p\ge 1,$ and the truncation error is bounded by

$$\begin{aligned} \Vert Z - Z^\kappa \Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))} \le {\hat{C}}_p \cdot \kappa ^{-(\alpha -2)/2} \end{aligned}$$

for $\kappa > \ell _0,$ where ${\hat{C}}_p$ is a constant depending on p,  C,  and $\alpha .$

In addition, $(Z^\kappa , \kappa \in {\mathbb {N}})$ converges ${\mathbb {P}}$-almost surely and for all $\delta < (\alpha -2)/2,$ the truncation error is asymptotically bounded by

$$\begin{aligned} \Vert Z - Z^\kappa \Vert _{L^2({\mathbb {S}}^2)} \le \kappa ^{-\delta }, \quad {\mathbb {P}}\text{-a.s.} \end{aligned}$$

That is, for almost all $\omega \in \varOmega ,$ there exists $\kappa _0(\omega )$ such that for all $\kappa >\kappa _0(\omega ),$ $\Vert Z(\omega ) - Z^\kappa (\omega )\Vert _{L^2({\mathbb {S}}^2)} \le \kappa ^{-\delta }$ is satisfied.

We follow [32], where isotropic Gaussian random fields are connected to Q-Wiener processes. There it is shown that an isotropic Q-Wiener process $(W(t), t \in {\mathbb {T}})$ on some finite time interval ${\mathbb {T}}= [0,T]$ with values in $L^2({\mathbb {S}}^2)$ can be represented by the expansion

$$\begin{aligned} \begin{aligned}&W(t,y)\\ &\quad = \sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell a^{\ell , m}(t) Y_{\ell , m}(y) \\ &\quad = \sum _{\ell =0}^\infty \left( \sqrt{A_\ell } \beta _1^{\ell , 0}(t) Y_{\ell , 0}(y) + \sqrt{2A_\ell } \sum _{m=1}^\ell (\beta _1^{\ell , m}(t) \text{Re}\, Y_{\ell , m}(y) + \beta _2^{\ell , m}(t) \text{Im}\, Y_{\ell , m}(y))\right) \\ &\quad = \sum _{\ell =0}^\infty \left( \sqrt{A_\ell } \beta _1^{\ell , 0}(t) L_{\ell , 0}(\vartheta ) + \sqrt{2A_\ell } \sum _{m=1}^\ell L_{\ell , m}(\vartheta ) (\beta _1^{\ell , m}(t) \cos (m\varphi ) + \beta _2^{\ell , m}(t) \sin (m \varphi ))\right) , \end{aligned} \end{aligned}$$

(2)

where $(\beta _i^{\ell , m}, i=1,2, \ell \in {\mathbb {N}}_0, m=0,\ldots , \ell )$ is a sequence of independent, real-valued Brownian motions with $\beta _2^{\ell , 0} = 0$ for $\ell \in {\mathbb {N}}_0$ and $t \in {\mathbb {T}}.$ The covariance operator Q is characterized similarly to the introduction in [31] by

$$\begin{aligned} Q Y_{\ell , m}&= A_\ell Y_{\ell , m} \end{aligned}$$

for $\ell \in {\mathbb {N}}_0$ and $m=-\ell ,\ldots ,\ell ,$ i.e., the eigenvalues of Q are given by the angular power spectrum $(A_\ell , \ell \in {\mathbb {N}}_0),$ and the eigenfunctions are the spherical harmonic functions.

Due to the properties of Brownian motion, the above Q-Wiener process can be generated by increments which are isotropic Gaussian random fields with angular power spectrum $(hA_\ell , \ell \in {\mathbb {N}}_0)$ for a time step size h.

3 The stochastic wave equation on the sphere

With the preparations from the preceding section at hand, we have all necessary tools to introduce the main subject of our study.

The stochastic wave equation on the sphere is defined as

$$\begin{aligned} \partial _{tt}u(t)-\varDelta _{{\mathbb {S}}^2}u(t)={\dot{W}}(t) \end{aligned}$$

(3)

with initial conditions $u(0)=v_1\in L^2(\varOmega ;L^2({\mathbb {S}}^2))$ and $\partial _{t}u(0)=v_2\in L^2(\varOmega ;L^2({\mathbb {S}}^2)),$ where $t \in {\mathbb {T}}= [0,T],$ $T < + \infty .$ For ease of presentation, from now on, we consider the case of non-random initial conditions. The case of random initial conditions follows under appropriate integrability assumptions. The notation ${\dot{W}}$ stands for the formal derivative of the Q-Wiener process with series expansion (2) as introduced in Sect. 2.

Denoting the velocity of the solution by $u_2 = \partial _{t} u_1 = \partial _{t} u,$ one can rewrite (3) as

$$\begin{aligned} \mathop {}\!\mathrm {d}X(t)&=AX(t)\,\mathop {}\!\mathrm {d}t+G\,\mathop {}\!\mathrm {d}W(t) \\ X(0)&=X_0, \end{aligned}$$

(4)

where

$$\begin{aligned} A=\left( \begin{array}{cc} 0 &{} I \\ \varDelta _{{\mathbb {S}}^2} &{} 0 \end{array}\right) , \quad G=\left( \begin{array}{c} 0\\ I \end{array}\right) , \quad X=\left( \begin{array}{c} u_1\\ u_2\end{array}\right) , \quad X_0=\left( \begin{array}{c} v_1\\ v_2\end{array}\right) . \end{aligned}$$

Existence of a unique mild solution of the abstract formulation (4) of the stochastic wave equation on the sphere follows from classical results on linear SPDEs, see for instance [12], and this mild solution reads

$$\begin{aligned} X(t)=\mathrm {e}^{tA}X_0+\int _0^t\mathrm {e}^{(t-s)A}G\,\mathop {}\!\mathrm {d}W(s). \end{aligned}$$

Equivalently, the integral formulation of our problem is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} u_1(t)&{}=v_1+\int _0^tu_2(s)\,\mathop {}\!\mathrm {d}s\\ u_2(t)&{}=v_2+\int _0^t\varDelta _{{\mathbb {S}}^2}u_1(s)\,\mathop {}\!\mathrm {d}s+W(t). \end{array}\right. } \end{aligned}$$

(5)

Since the spherical harmonic functions ${\mathscr {Y}}=(Y_{\ell ,m}, \ell \in {\mathbb {N}}_0, m=-\ell ,\ldots ,\ell )$ form an orthonormal basis of $L^2({\mathbb {S}}^2)$ and are eigenfunctions of $\varDelta _{{\mathbb {S}}^2},$ we insert the following ansatz for a series expansion of the exact solution to SPDE (3)

$$\begin{aligned} u_1(t)=\sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell u_1^{\ell ,m}(t)Y_{\ell , m}\quad \text{and}\quad u_2(t)=\sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell u_2^{\ell ,m}(t)Y_{\ell , m} \end{aligned}$$

(6)

into equation (5) and compare the coefficients in front of $Y_{\ell ,m}$ to obtain the following system

$$\begin{aligned} u_1^{\ell ,m}(t)&=v_1^{\ell ,m}+\int _0^t u_2^{\ell ,m}(s)\,\mathop {}\!\mathrm {d}s\\ u_2^{\ell ,m}(t)&=v_2^{\ell ,m}-\ell (\ell +1)\int _0^t u_1^{\ell ,m}(s)\,\mathop {}\!\mathrm {d}s+a^{\ell ,m}(t), \end{aligned}$$

where $v_1^{\ell ,m},$ $v_2^{\ell ,m},$ resp. $a^{\ell ,m}$ are the coefficients of the expansions of the initial values $v_1$ and $v_2,$ resp. weighted Brownian motions in the expansion of the noise (2). The coefficients $a^{\ell ,m}$ are given by $a^{\ell ,m}(t)=(W(t),Y_{\ell ,m})_{L^2({\mathbb {S}}^2)},$ see Lemma 1. More precise relations between the coefficients $a^{\ell ,m}$ and the angular power spectrum $A_\ell$ can be derived from the proof of [32, Lemma 5.1].

Writing the evolution of the initial values in the above linear harmonic oscillators with rotation matrices and using the variation of constants formula, one derives the following system for the coefficients of the expansions of the solution

$$\begin{aligned} {\left\{ \begin{array}{ll} u_1^{\ell ,m}(t)&{}=\cos (t(\ell (\ell +1))^{1/2})v_1^{\ell ,m}+(\ell (\ell +1))^{-1/2}\sin (t(\ell (\ell +1))^{1/2})v_2^{\ell ,m}+{{\hat{W}}}_1^{\ell ,m}(t)\\ u_2^{\ell ,m}(t)&{}=-(\ell (\ell +1))^{1/2}\sin (t(\ell (\ell +1))^{1/2})v_1^{\ell ,m}+\cos (t(\ell (\ell +1))^{1/2})v_2^{\ell ,m}+{{\hat{W}}}_2^{\ell ,m}(t), \end{array}\right. } \end{aligned}$$

(7)

where

$$\begin{aligned} {{\hat{W}}}^{\ell ,m}(t) = \left( \begin{array}{c} {{\hat{W}}}_1^{\ell ,m}(t)\\ {{\hat{W}}}_2^{\ell ,m}(t) \end{array}\right) =\int _0^t R^\ell (t-s) \,\mathop {}\!\mathrm {d}a^{\ell ,m}(s) \end{aligned}$$

with

$$\begin{aligned} R^\ell (t) = \left( \begin{array}{c} R^\ell _1(t) \\ R^\ell _2(t) \end{array}\right) = \left( \begin{array}{c} (\ell (\ell +1))^{-1/2}\sin (t(\ell (\ell +1))^{1/2})\\ \cos (t(\ell (\ell +1))^{1/2}) \end{array}\right) \end{aligned}$$

for $\ell \ne 0$ and

$$\begin{aligned} {{\hat{W}}}^{0,0}(t) = \left( \begin{array}{c} {{\hat{W}}}_1^{0,0}(t)\\ {{\hat{W}}}_2^{0,0}(t) \end{array}\right) = \left( \begin{array}{c} \int _0^t a^{0,0}(s) \, \mathop {}\!\mathrm {d}s\\ a^{0,0}(t) \end{array}\right) . \end{aligned}$$

Coming back to the above mild solution and using the definition of ${{\hat{W}}}^{\ell ,m},$ we observe that, for $t>s,$

$$\begin{aligned} X(t)&=\mathrm {e}^{(t-s)A}X(s)+\int _s^t\mathrm {e}^{(t-r)A}G\,\mathop {}\!\mathrm {d}W(r)=\mathrm {e}^{(t-s)A}X(s)\\ &\quad +\int _0^{t-s}\mathrm {e}^{(t-s-r)A}G\,\mathop {}\!\mathrm {d}W(s+r)\\ &=\mathrm {e}^{(t-s)A}X(s)+{{\hat{W}}}(t)-\mathrm {e}^{(t-s)A}{{\hat{W}}}(s), \end{aligned}$$

with the notation

$$\begin{aligned} {{\hat{W}}}(t,y)= \left( \begin{array}{c} {{\hat{W}}}_1(t,y)\\ {{\hat{W}}}_2(t,y) \end{array}\right) = \sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell {{\hat{W}}}^{\ell , m}(t) Y_{\ell , m}(y). \end{aligned}$$

With a slight abuse, we call the quantity ${{\hat{W}}}(t)-\mathrm {e}^{(t-s)A}{{\hat{W}}}(s)$ an increment.

We now characterize the above stochastic convolution ${{\hat{W}}}$ in the following proposition.

Proposition 1

For all $t>0,$ the stochastic convolution ${{\hat{W}}}(t)$ is Gaussian with mean zero and expansion

$$\begin{aligned}&{{\hat{W}}}(t,y)\\ &\quad = \sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell {{\hat{W}}}^{\ell , m}(t) Y_{\ell , m}(y)\\ &\quad = \sum _{\ell =0}^\infty \left( \sqrt{A_\ell } {{\hat{\beta }}}_1^{\ell , 0}(t) Y_{\ell , 0}(y) + \sqrt{2A_\ell } \sum _{m=1}^\ell (\hat{\beta }_1^{\ell , m}(t) \text{Re}\, Y_{\ell , m}(y) + {{\hat{\beta }}}_2^{\ell , m}(t) \text{Im}\, Y_{\ell , m}(y))\right) \\ &\quad = \sum _{\ell =0}^\infty \left( \sqrt{A_\ell } \hat{\beta }_1^{\ell , 0}(t) L_{\ell , 0}(\vartheta ) + \sqrt{2A_\ell } \sum _{m=1}^\ell L_{\ell , m}(\vartheta ) ({{\hat{\beta }}}_1^{\ell , m}(t) \cos (m\varphi ) + \hat{\beta }_2^{\ell , m}(t) \sin (m \varphi ))\right) , \end{aligned}$$

where equality is in distribution.

The processes $({{\hat{\beta }}}^{\ell ,m}_i(t), i=1,2, \ell \in {\mathbb {N}}_0, m=-\ell ,\ldots , \ell )$ are given by

$$\begin{aligned} {{\hat{\beta }}}^{\ell ,m}_i(t) = \left( \begin{array}{c} {\hat{\beta }}_{i,1}^{\ell ,m}(t)\\ {\hat{\beta }}_{i,2}^{\ell ,m}(t) \end{array}\right) = D_\ell (t) X^{\ell ,m}_i \end{aligned}$$

for a sequence $(X^{\ell ,m}_i = (X_{i,1}^{\ell ,m},X_{i,2}^{\ell ,m})^T, i=1,2, \ell \in {\mathbb {N}}_0, m = -\ell , \ldots , \ell )$ of independent, identically distributed random variables with $X^{\ell ,m}_{i,j} \sim {\mathscr {N}}\,(0,1).$ The term $D_\ell (t)$ denotes the Cholesky decomposition of the covariance matrix $C_\ell (t)$ of ${{\hat{W}}}^{\ell ,m}(t).$ More specifically, $D_\ell$ satisfies

$$\begin{aligned} D_\ell (t) D_\ell (t)^T = C_\ell (t) \end{aligned}$$

with

$$\begin{aligned} C_\ell (t) = \left( \begin{array}{cc} \frac{2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t)}{4(\ell (\ell +1))^{3/2}} &{} \frac{\sin ((\ell (\ell +1))^{1/2}t)^2}{2(\ell (\ell +1))}\\ \frac{\sin ((\ell (\ell +1))^{1/2}t)^2}{2(\ell (\ell +1))} &{} \frac{2(\ell (\ell +1))^{1/2}t+\sin (2(\ell (\ell +1))^{1/2}t)}{4(\ell (\ell +1))^{1/2}} \end{array}\right) \end{aligned}$$

for $\ell \ne 0$ and

$$\begin{aligned} C_0(t) = \left( \begin{array}{cc} t^3/3 &{} t^2/2\\ t^2/2 &{} t \end{array}\right) . \end{aligned}$$

Proof

We observe first that ${{\hat{W}}}(t)$ satisfies by (2)

$$\begin{aligned} {{\hat{W}}}(t)&= \sum _{\ell = 0}^\infty \sum _{m=-\ell }^\ell {{\hat{W}}}^{\ell ,m}(t) Y_{\ell ,m}\\ &= {{\hat{W}}}^{0,0}(t) Y_{0,0} + \sum _{\ell = 1}^\infty \sum _{m=-\ell }^\ell \int _0^t R^\ell (t-s) \,\mathop {}\!\mathrm {d}a^{\ell ,m}(s) Y_{\ell ,m}\\ &= {{\hat{W}}}^{0,0}(t) Y_{0,0} + \sum _{\ell = 1}^\infty \sqrt{A_\ell } \left[ \int _0^t R^\ell (t-s) \, \mathop {}\!\mathrm {d}\beta _1^{\ell ,0}(s) Y_{\ell ,0}\right. \\ &\quad\left. + \sqrt{2} \sum _{m=1}^\ell \left( \int _0^t R^\ell (t-s) \, \mathop {}\!\mathrm {d}\beta _1^{\ell ,m}(s) \text{Re}\, Y_{\ell ,m} + \int _0^t R^\ell (t-s) \, \mathop {}\!\mathrm {d}\beta _2^{\ell ,m}(s) \text{Im}\, Y_{\ell ,m} \right) \right] \end{aligned}$$

with Brownian motions $(\beta _1^{\ell ,m}, \ell \in {\mathbb {N}}_0, m=0,\ldots ,\ell )$ and $(\beta _2^{\ell ,m}, \ell \in {\mathbb {N}}, m=1,\ldots ,\ell )$ which are independent. Since all Brownian motions are centered and independent, it is sufficient to compute the following covariances which are given by

$$\begin{aligned} C_0(t)&=A_0^{-1} \left( \begin{array}{cc} {{\,\mathrm{{\mathbb {E}}}\,}}[{{\hat{W}}}^{0,0}_1(t)^2] &{} {{\,\mathrm{{\mathbb {E}}}\,}}[{{\hat{W}}}_1^{0,0}(t){{\hat{W}}}_2^{0,0}(t)]\\ {{\,\mathrm{{\mathbb {E}}}\,}}[{{\hat{W}}}_1^{0,0}(t){{\hat{W}}}_2^{0,0}(t)] &{} {{\,\mathrm{{\mathbb {E}}}\,}}[{{\hat{W}}}_2^{0,0}(t)^2] \end{array}\right) \\ &=A_0^{-1} \left( \begin{array}{cc} {{\,\mathrm{{\mathbb {E}}}\,}}[(\int _0^t a^{0,0}(s) \, \mathop {}\!\mathrm {d}s)^2] &{} {{\,\mathrm{{\mathbb {E}}}\,}}[ \int _0^t a^{0,0}(s) a^{0,0}(t) \, \mathop {}\!\mathrm {d}s]\\ {{\,\mathrm{{\mathbb {E}}}\,}}[ \int _0^t a^{0,0}(s) a^{0,0}(t) \, \mathop {}\!\mathrm {d}s] &{} {{\,\mathrm{{\mathbb {E}}}\,}}[(a^{0,0}(t))^2] \end{array}\right) = \left( \begin{array}{cc} t^3/3 &{} t^2/2\\ t^2/2 &{} t \end{array}\right) \end{aligned}$$

for $\ell = 0$ and else for $i=1,2$

$$\begin{aligned} C_\ell (t)&= \left( \begin{array}{cc} {{\,\mathrm{{\mathbb {E}}}\,}}[(\text{Int}_1)^2] &{} {{\,\mathrm{{\mathbb {E}}}\,}}[\text{Int}_1\text{Int}_2]\\ {{\,\mathrm{{\mathbb {E}}}\,}}[\text{Int}_1 \text{Int}_2] &{} {{\,\mathrm{{\mathbb {E}}}\,}}[(\text{Int}_2)^2] \end{array}\right) \\ &= \left( \begin{array}{cc} \int _0^t R^\ell _1(t-s)^2 \, \mathop {}\!\mathrm {d}s &{} \int _0^t R^\ell _1(t-s) R^\ell _2(t-s) \, \mathop {}\!\mathrm {d}s\\ \int _0^t R^\ell _1(t-s) R^\ell _2(t-s) \, \mathop {}\!\mathrm {d}s &{} \int _0^t R^\ell _2(t-s)^2 \, \mathop {}\!\mathrm {d}s \end{array}\right) \\ &= \left( \begin{array}{cc} \frac{2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t)}{4(\ell (\ell +1))^{3/2}} &{} \frac{\sin ((\ell (\ell +1))^{1/2}t)^2}{2(\ell (\ell +1))}\\ \frac{\sin ((\ell (\ell +1))^{1/2}t)^2}{2(\ell (\ell +1))} &{} \frac{2(\ell (\ell +1))^{1/2}t+\sin (2(\ell (\ell +1))^{1/2}t)}{4(\ell (\ell +1))^{1/2}} \end{array}\right) , \end{aligned}$$

where we have set $\text{Int}_1=\int _0^t R^\ell _1(t-s) \, \mathop {}\!\mathrm {d}\beta _i^{\ell ,m}(s)$ and $\text{Int}_2=\int _0^t R^\ell _2(t-s) \, \mathop {}\!\mathrm {d}\beta _i^{\ell ,m}(s)$ and used Itô’s isometry.

Setting $D_\ell (t)$ the Cholesky decomposition of the above covariance matrices satisfying

$$\begin{aligned} D_\ell (t)^T D_\ell (t) = C_\ell (t) \end{aligned}$$

we obtain for $\ell \ne 0$ and $i=1,2$ that

$$\begin{aligned} D_\ell (t)X^{\ell ,m}_i = \int _0^t R^\ell (t-s) \, \mathop {}\!\mathrm {d}\beta _i^{\ell ,m}(s) \end{aligned}$$

in distribution and similarly for $\ell = 0$

$$\begin{aligned} D_\ell (t)X^{\ell ,m}_1 = {{\hat{W}}}^{0,0}(t) \end{aligned}$$

with $(X^{\ell ,m}_i = (X_{i,1}^{\ell ,m},X_{i,2}^{\ell ,m})^T, i=1,2, \ell \in {\mathbb {N}}_0, m = -\ell , \ldots , \ell )$ independent and identically distributed standard normally distributed random variables. This concludes the proof. $\square$

Since we are interested in the simulation of sample paths of solutions to (4), we need to generate increments of ${{\hat{W}}}^{\ell ,m}.$ Therefore it is important to observe that

$$\begin{aligned} {{\hat{\beta }}}^{\ell ,m}_i(t) - {{\hat{\beta }}}^{\ell ,m}_i(s) = D_\ell (t-s) X^{\ell ,m}_i \end{aligned}$$

in distribution for $s<t.$ In this way we can generate sample paths of ${{\hat{W}}}$ by sums of independent Gaussian increments, see Algorithm 1. We now characterize the properties of these increments.

Corollary 1

For $0\le s_1<t_1\le s_2<t_2,$ the increments satisfy

• 1. ${{\hat{W}}}(t_1)-\exp ((t_1-s_1)A){{\hat{W}}}(s_1)$ is Gaussian,

• 2. ${{\hat{W}}}(t_1)-\exp ((t_1-s_1)A){{\hat{W}}}(s_1)= {{\hat{W}}}(t_1-s_1)$ in distribution,

• 3. ${{\hat{W}}}(t_1)-\exp ((t_1-s_1)A){{\hat{W}}}(s_1)$ and ${{\hat{W}}}(t_2)-\exp ((t_2-s_2)A){{\hat{W}}}(s_2)$ are independent.

The proof is a consequence of Proposition 1.

Remark 1

For completeness we also remark that the Cholesky decomposition $D_\ell (t)$ can be computed explicitly and is given by

$$\begin{aligned} D_\ell (t) = \left( \begin{array}{cc} d_{1,1} &{} d_{1,2}\\ 0 &{} d_{2,2} \end{array}\right) \end{aligned}$$

(8)

with

$$\begin{aligned} d_{1,1}&= \frac{(2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t))^{1/2}}{2(\ell (\ell +1))^{3/4}}\\ d_{1,2}&= \frac{\sin ((\ell (\ell +1))^{1/2}t)^2}{(\ell (\ell +1))^{1/4}(2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t))^{1/2}}\\ d_{2,2}&= \left( \frac{4(\ell (\ell +1))t^2 - \sin (2(\ell (\ell +1))^{1/2}t)^2 - 4\sin ((\ell (\ell +1))^{1/2}t)^4}{4(\ell (\ell +1))^{1/2}(2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t))} \right) ^{1/2} \end{aligned}$$

for $\ell \ne 0$ and

$$\begin{aligned} D_0(t) = t^{1/2} \left( \begin{array}{cc} t/\sqrt{3} &{} \sqrt{3}/2\\ 0 &{} 1/2 \end{array}\right) . \end{aligned}$$

We close this section by showing regularity estimates for the solution of (3) that depend on the regularity of the initial conditions and the driving noise. These properties allow to obtain optimal weak convergence rates in Sect. 4.

Proposition 2

Denote by $X=(u_1,u_2)$ the solution to the stochastic wave equation (4) with initial value $(v_1,v_2).$ Assume that there exist $\ell _0 \in {\mathbb {N}},$ $\alpha > 2,$ and a constant $C>0$ such that the angular power spectrum of the driving noise $(A_\ell , \ell \in {\mathbb {N}}_0)$ satisfies $A_\ell \le C \cdot \ell ^{-\alpha }$ for all $\ell > \ell _0.$ Then, for all $t \in [0,T],$ $q \in {\mathbb {N}},$ and $s < \alpha /2$ with $v_1 \in H^s({\mathbb {S}}^2)$ and $v_2 \in H^{s-1}({\mathbb {S}}^2),$ the solution satisfies $u_1(t) \in L^{2q}(\varOmega ;H^s({\mathbb {S}}^2)),$ i.e., there exists a constant $M_T$ such that

$$\begin{aligned} \Vert u_1(t)\Vert _{L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))} \le M_T (1 + \Vert v_1\Vert _{H^s({\mathbb {S}}^2)} + \Vert v_2\Vert _{H^{s-1}({\mathbb {S}}^2)}) < + \infty . \end{aligned}$$

And for all $t \in [0,T],$ $q \in {\mathbb {N}},$ and $s < \alpha /2 - 1$ with $v_1 \in H^{s+1}({\mathbb {S}}^2)$ and $v_2 \in H^s({\mathbb {S}}^2),$ $u_2(t) \in L^{2q}(\varOmega ;H^s({\mathbb {S}}^2)),$ i.e., there exists a constant $M_T$ such that

$$\begin{aligned} \Vert u_2(t)\Vert _{L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))} \le M_T (1 + \Vert v_1\Vert _{H^{s+1}({\mathbb {S}}^2)} + \Vert v_2\Vert _{H^s({\mathbb {S}}^2)}) < + \infty . \end{aligned}$$

Proof

Let us first observe that, using the definition of $u_1(t),$ see Eqs. (6) and (7), one has

$$\begin{aligned}&\Vert u_1(t)\Vert _{L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))}\\ &\le \left\| \sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell \left( R^\ell _2(t) v_1^{\ell ,m} +R^\ell _1(t) v_2^{\ell ,m}\right) Y_{\ell ,m}\right\| _{L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))}\\ &\quad +\left\| {\hat{W}}_1(t)\right\| _{L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))}.\\ \end{aligned}$$

Using the definitions of $R^\ell _1(t),$ $R^\ell _2(t),$ and of the $H^s$-norm, the first term with respect to the initial conditions satisfies

$$\begin{aligned}&\left\| \sum _{\ell =1}^\infty \sum _{m=-\ell }^\ell \left( R^\ell _2(t) v_1^{\ell ,m} +R^\ell _1(t) v_2^{\ell ,m}\right) Y_{\ell ,m}\right\| _{H^s({\mathbb {S}}^2)}^2\\ &\qquad \le 2 \sum _{\ell =1}^\infty \sum _{m=-\ell }^\ell \left( (1+\ell (\ell +1))^s |v_1^{\ell ,m}|^2 + (1+\ell (\ell +1))^s (\ell (\ell +1))^{-1} |v_2^{\ell ,m}|^2 \right) \\ &\qquad \le C (\Vert v_1\Vert _{H^s({\mathbb {S}}^2)}^2 + \Vert v_2\Vert _{H^{s-1}({\mathbb {S}}^2)}^2). \end{aligned}$$

Given the angular power spectrum of ${\hat{W}}(t)$ in Proposition 1, it follows for the second moment, i.e. $q=1,$ that

$$\begin{aligned} \left\| {\hat{W}}_1(t)\right\| _{L^{2}(\varOmega ;H^s({\mathbb {S}}^2))}^2&= \left\| (\text{Id}-\varDelta _{{\mathbb {S}}^2})^{s/2} {\hat{W}}_1(t)\right\| _{L^{2}(\varOmega ;L^2({\mathbb {S}}^2))}^2\\ &= \sum _{\ell = 0}^\infty (2\ell +1)(1+\ell (\ell +1))^s\\ &\quad \times \frac{2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t)}{4(\ell (\ell +1))^{3/2}}A_\ell , \end{aligned}$$

which converges for $\alpha > 2s$ since the elements of the sum behave like $\ell ^{2s-\alpha -1}.$ By Fernique’s theorem [16], this convergence implies that the norm is finite for all q and arbitrary moment bounds can for example be obtained by the Burkholder–Davis–Gundy inequality, see [12, Section 4.6] for instance.

Similar computations for $u_2$ conclude the proof. $\square$

4 Convergence analysis

In this section, we numerically solve the wave equation driven by additive Q-Wiener noise with spectral methods. We approximate the solution by truncation of the derived spectral representation and show convergence rates in p-th moment, ${\mathbb {P}}$-almost surely, and in the weak sense.

An efficient simulation of numerical approximations to solutions to the stochastic wave equation on the sphere (3) is then obtained via Algorithm 1. Observe that the stochastic integrals in the proposed algorithm are not discretized, see the previous section. Hence, the time integration is done exactly on any uniform grid on a finite time interval [0, T]. This allows to generate sample paths of numerical solutions with any chosen time resolution that do not suffer from an additional error in the time discretization.

The strong errors of this truncation procedure in Algorithm 1 are given in the following proposition.

Proposition 3

Let $t \in {\mathbb {T}}$ and $0=t_0< \cdots < t_n = t$ be a discrete time partition for $n \in {\mathbb {N}},$ which yields a recursive representation of the solution $X=(u_1,u_2)$ of the stochastic wave equation on the sphere (4) given by (6). Assume that the initial values satisfy $v_1\in H^\beta ({\mathbb {S}}^2)$ and $v_2\in H^\gamma ({\mathbb {S}}^2)$ for some $\beta >0$ and $\gamma >1.$ Furthermore, assume that there exist $\ell _0 \in {\mathbb {N}},$ $\alpha > 2,$ and a constant $C>0$ such that the angular power spectrum of the driving noise $(A_\ell , \ell \in {\mathbb {N}}_0)$ satisfies $A_\ell \le C \cdot \ell ^{-\alpha }$ for all $\ell > \ell _0.$ Then, the error of the approximate solution $X^\kappa =(u_1^\kappa ,u_2^\kappa ),$ given by (9), is bounded uniformly by

$$\begin{aligned} \Vert u_1(t) - u_1^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}&\le {\hat{C}}_p \cdot \left( \kappa ^{-\alpha /2}+\kappa ^{-\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)}+ \kappa ^{-(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \\ \Vert u_2(t) - u_2^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}&\le {\hat{C}}_p \cdot \left( \kappa ^{-(\alpha /2-1)}+\kappa ^{-(\beta -1)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)}+ \kappa ^{-\gamma }\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \end{aligned}$$

for all $p\ge 1$ and $\kappa > \ell _0,$ where ${\hat{C}}_p$ is a constant that may depend on p,  C,  T,  and $\alpha$ but is independent of n,  h,  t.

On top of that, the error of the approximate solution $X^\kappa$ is bounded uniformly and asymptotically in $\kappa$ by

$$\begin{aligned} \Vert u_1(t) - u_1^\kappa (t)\Vert _{L^2({\mathbb {S}}^2)}&\le \kappa ^{-\delta }, \quad {\mathbb {P}}\text{-a.s.} \\ \Vert u_2(t) - u_2^\kappa (t)\Vert _{L^2({\mathbb {S}}^2)}&\le \kappa ^{-(\delta -1)}, \quad {\mathbb {P}}\text{-a.s.} \end{aligned}$$

for all $\delta < \min (\alpha /2,\beta ,\gamma +1).$

Remark 2

We remark that it is not necessary that the angular power spectrum $(A_\ell , \ell \in {\mathbb {N}}_0)$ of the Q-Wiener process decays with rate $\ell ^{-\alpha }$ for $\alpha >2$ but that it is sufficient to assume that $\alpha > 0$ to show convergence in the first component, see the numerical experiment in Sect. 5. I.e., we do not require that Q is a trace class operator for convergence in the first component.

Proof of Proposition 3

Let us first consider the convergence in p-th moment of the first component of the solution for $p\ge 1.$ By definition of $u_1$ and $u_1^\kappa$ in (6) and (9), one obtains

$$\begin{aligned}&\left\| u_1(t)-u_1^\kappa (t)\right\| _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}\\ &\quad = \left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell u_1^{\ell ,m}(t)Y_{\ell ,m}\right\| _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}\\ &\quad \le \left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell \left( R^\ell _2(t) v_1^{\ell ,m} +R^\ell _1(t) v_2^{\ell ,m}\right) Y_{\ell ,m}\right\| _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}\\ &\qquad +\left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell \int _0^t R^\ell _1(t-s)\,\mathop {}\!\mathrm {d}a^{\ell ,m}(s)Y_{\ell ,m}\right\| _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}\\ &\quad \le \Vert v_1 - v_1^\kappa \Vert _{L^2({\mathbb {S}}^2)} + \left( \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell (\ell (\ell +1))^{-1}|v_2^{\ell ,m}|^2\right) ^{1/2}\\ &\qquad + \Vert Z-Z^\kappa \Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}, \end{aligned}$$

where Z denotes an isotropic Gaussian random field with angular power spectrum

$$\begin{aligned} \widetilde{A}_\ell =\frac{2(\ell (\ell +1))^{1/2}t-\sin (2(\ell (\ell +1))^{1/2}t)}{4(\ell (\ell +1))^{3/2}}A_\ell \end{aligned}$$

and $Z^\kappa$ its truncation. Similarly, the notation $v_1^\kappa$ is used for the truncation of the initial value $v_1.$ Observe that

$$\begin{aligned} {{\widetilde{A}}}_\ell \le C\left( \frac{t}{2\ell (\ell +1)}+\frac{|\sin (2(\ell (\ell +1))^{1/2}t)|}{4(\ell (\ell +1))^{3/2}}\right) A_\ell \le C T \ell ^{-2}A_\ell \le C T \ell ^{-(\alpha +2)} \end{aligned}$$

for large enough index $\ell > \ell _0 \ge 0$ using the assumption on the angular power spectrum of the noise $A_\ell .$ The assumptions on the initial values and the fact that the spherical harmonic functions are orthonormal provide us with the estimate

$$\begin{aligned} \left\| v_1 - v_1^\kappa \right\| _{L^2({\mathbb {S}}^2)}&=\left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell v_1^{\ell ,m}Y_{\ell ,m}\right\| _{L^2({\mathbb {S}}^2)} \\ &=\left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell v_1^{\ell ,m}(\text{Id}-\varDelta _{{\mathbb {S}}^2})^{\beta /2} (\text{Id}-\varDelta _{{\mathbb {S}}^2})^{-\beta /2}Y_{\ell ,m}\right\| _{L^2({\mathbb {S}}^2)}\\ &=\left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell v_1^{\ell ,m}(1+\ell (\ell +1))^{-\beta /2}(\text{Id}-\varDelta _{{\mathbb {S}}^2})^{\beta /2} Y_{\ell ,m}\right\| _{L^2({\mathbb {S}}^2)}\\ &\le C \kappa ^{-\beta }\left\| v_1\right\| _{H^\beta ({\mathbb {S}}^2)} \end{aligned}$$

and similarly for the second component of the initial value.

Collecting all the estimates above and using Theorem 1 we obtain the desired bound

$$\begin{aligned} \Vert u_1(t)-u_1^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))} \le {{\hat{C}}}_p\cdot \left( \kappa ^{-\alpha /2}+\kappa ^{-\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} + \kappa ^{-(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)}\right) . \end{aligned}$$

The corresponding estimate for the second component is done in a similar way and left to the reader. Observe that the rate of convergence decays by one due to the factor $(\ell (\ell +1))^{1/2}$ in the first term of (7).

We continue with the rate of the almost sure convergence in the first component of the solution. Let $\delta < \min (\alpha /2,\beta ,\gamma +1).$ The above strong $L^p$ error estimate combined with Chebyshev’s inequality provide us with

$$\begin{aligned}&{\mathbb {P}}\left( \Vert u_1(t)-u_1^\kappa (t)\Vert _{L^2({\mathbb {S}}^2)} \ge \kappa ^{-\delta }\right) \\ &\quad \le \kappa ^{\delta p} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert u_1(t)-u_1^\kappa (t)\Vert _{L^2({\mathbb {S}}^2)}^p\right] \\ &\quad \le \kappa ^{\delta p}{{\hat{C}}}_p^p\left( \kappa ^{-\alpha /2}+\kappa ^{-\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)}+ \kappa ^{-(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)}\right) ^p. \end{aligned}$$

For all $p > \max ( (\alpha /2-\delta )^{-1}, (\beta -\delta )^{-1}, (\gamma +1-\delta )^{-1} ),$ the series

$$\begin{aligned} \sum _{\kappa =1}^\infty \kappa ^{(\delta -\min (\alpha /2,\beta ,\gamma +1))p} < + \infty \end{aligned}$$

converges which implies the claim by the Borel–Cantelli lemma. Almost sure convergence of the second component is shown in a similar way which concludes the proof. $\square$

Using Proposition 3, we continue with bounding weak errors of the mean and second moment in a first step. We observe that the weak error for the mean is the error to the corresponding deterministic wave equation on ${\mathbb {S}}^2$ and that the error for the second moment satisfies the rule of thumb that the weak convergence rate is twice the strong convergence rate.

Proposition 4

Let $t \in {\mathbb {T}}$ and $0=t_0< \cdots < t_n = t$ be a discrete time partition for $n \in {\mathbb {N}}$ which yields a recursive representation of the solution $X=(u_1,u_2)$ of the stochastic wave equation on the sphere (4) given by (6). Assume that the initial values satisfy $v_1\in H^\beta ({\mathbb {S}}^2)$ and $v_2\in H^\gamma ({\mathbb {S}}^2).$ Furthermore, assume that there exist $\ell _0 \in {\mathbb {N}},$ $\alpha > 2,$ and a constant $C>0$ such that the angular power spectrum of the driving noise $(A_\ell , \ell \in {\mathbb {N}}_0)$ satisfies $A_\ell \le C \cdot \ell ^{-\alpha }$ for all $\ell > \ell _0.$

Then, the errors in mean of the approximate solution $X^\kappa =(u_1^\kappa ,u_2^\kappa ),$ given by (9), are bounded uniformly by

$$\begin{aligned} \left\| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ u_1(t) - u_1^\kappa (t)\right] \right\| _{L^2({\mathbb {S}}^2)}&\le {\hat{C}} \cdot \left( \kappa ^{-\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} + \kappa ^{-(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \\ \left\| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ u_2(t) - u_2^\kappa (t)\right] \right\| _{L^2({\mathbb {S}}^2)}&\le {\hat{C}} \cdot \left( \kappa ^{-(\beta -1)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} +\kappa ^{-\gamma }\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \end{aligned}$$

for all $\kappa > \ell _0,$ where ${\hat{C}}$ is a constant that may depend on C,  T,  and $\alpha$ but is independent of n,  h,  t.

Furthermore, the errors of the second moment are bounded by

$$\begin{aligned}&\left| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert u_1(t)\Vert ^2_{L^2({\mathbb {S}}^2)} - \Vert u_1^\kappa (t)\Vert ^2_{L^2({\mathbb {S}}^2)}\right] \right| \\ &\quad \le {\hat{C}} \cdot \left( \kappa ^{-\alpha }+\kappa ^{-2\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} + \kappa ^{-2(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \\ &\left| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert u_2(t)\Vert ^2_{L^2({\mathbb {S}}^2)} - \Vert u_2^\kappa (t)\Vert ^2_{L^2({\mathbb {S}}^2)}\right] \right| \\ &\quad \le {\hat{C}} \cdot \left( \kappa ^{-(\alpha -2)}+\kappa ^{-2(\beta -1)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} + \kappa ^{-2\gamma }\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \end{aligned}$$

for all $\kappa > \ell _0,$ where ${\hat{C}}$ is a constant that may depend on C,  T,  and $\alpha$ but is independent of n,  h,  t.

Proof

The definition of X and its approximation $X^\kappa$ yield

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ u_j(t)-u_j^\kappa (t) \right] =\sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell {{\,\mathrm{{\mathbb {E}}}\,}}\left[ u_j^{\ell ,m}(t)\right] Y_{\ell ,m} \end{aligned}$$

for $j=1,2.$ Next, using (7) and the properties of ${{\hat{W}}}_1^{\ell ,m}(t)$ from Proposition 1, one obtains

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}\left[ u_1^{\ell ,m}(t)\right] =\cos (t(\ell (\ell +1))^{1/2})v_1^{\ell ,m} +(\ell (\ell +1))^{-1/2}\sin (t(\ell (\ell +1))^{1/2})v_2^{\ell ,m} \end{aligned}$$

and similarly for the second component. This corresponds to the errors in the initial values, see the proof of Proposition 3, and we thus obtain the error bound

$$\begin{aligned} \Vert {{\,\mathrm{{\mathbb {E}}}\,}}\left[ u_1(t) - u_1^\kappa (t)\right] \Vert _{L^2({\mathbb {S}}^2)} \le {\hat{C}} \cdot \left( \kappa ^{-\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} + \kappa ^{-(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \end{aligned}$$

and correspondingly for the second component.

In order to bound the second moments, we observe that

$$\begin{aligned}&{{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert u_j(t)\Vert ^2_{L^2({\mathbb {S}}^2)} - \Vert u_j^\kappa (t)\Vert ^2_{L^2({\mathbb {S}}^2)}\right] = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \langle u_j(t)+u_j^\kappa (t),u_j(t)-u_j^\kappa (t)\rangle _{L^2({\mathbb {S}}^2)} \right] \\ &\quad = {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \left\langle 2\sum _{\ell =0}^\kappa \sum _{m=-\ell }^\ell u_j^{\ell ,m}(t)Y_{\ell ,m}\right. \right. \\ &\qquad + \left. \left. \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell u_j^{\ell ,m}(t)Y_{\ell ,m}, \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell u_j^{\ell ,m}(t)Y_{\ell ,m} \right\rangle _{L^2({\mathbb {S}}^2)}\right] \\ &\quad =2{{\,\mathrm{{\mathbb {E}}}\,}}\left[ \sum _{\ell =0}^\kappa \sum _{m=-\ell }^\ell \sum _{\ell '=\kappa +1}^\infty \sum _{m'=-\ell '}^{\ell '} u_j^{\ell ,m}(t)u_j^{\ell ',m'}(t) \langle Y_{\ell ,m}, Y_{\ell ',m'}\rangle _{L^2({\mathbb {S}}^2)}\right] \\ &\qquad +{{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert u_j(t)-u_j^\kappa (t)\Vert ^2_{L^2({\mathbb {S}}^2)}\right] , \end{aligned}$$

for $j=1,2.$ Using the orthogonality of the spherical harmonics ${\mathscr {Y}},$ the first term vanishes and the second one is bounded by the square of the strong error in Proposition 3. This yields

$$\begin{aligned}&{{\,\mathrm{{\mathbb {E}}}\,}}\left[ \Vert u_1(t)\Vert ^2_{L^2({\mathbb {S}}^2)} - \Vert u_1^\kappa (t)\Vert ^2_{L^2({\mathbb {S}}^2)}\right] \\ &\quad \le {\hat{C}} \cdot \left( \kappa ^{-\alpha }+\kappa ^{-2\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)}+ \kappa ^{-2(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \end{aligned}$$

and similarly for the second component. $\square$

For a more general class of test functions, we obtain weak error rates that depend directly on the regularity of the test function and indirectly on the regularity of the solution. Let us first state the abstract assumption on the test functions that will be required for the next weak convergence result.

Assumption 1

For the SPDE (3) and some fixed $s>0,$ consider the class of Fréchet differentiable test functions $\varphi$ satisfying

$$\begin{aligned} \left\| \int _0^1\varphi '(\rho u_j(t)+(1-\rho )u_j^\kappa (t))\,\mathop {}\!\mathrm {d}\rho \right\| _{L^2(\varOmega ;H^s({\mathbb {S}}^2))} \le {\widetilde{C}} < + \infty \end{aligned}$$

for $j=1,2.$

This assumption characterizes the class of test functions by properties of the exact and numerical solutions to our problem. However, as seen next, this class includes test functions of polynomial growth, which do not explicitly depend on the exact and numerical solutions itself.

A typical example of a set of test functions satisfying the above assumption would be polynomial growth of the derivative in $H^s({\mathbb {S}}^2),$ i.e., to take $\varphi$ such that for all $x \in H^s({\mathbb {S}}^2)$

$$\begin{aligned} \Vert \varphi '(x)\Vert _{H^s({\mathbb {S}}^2)} \le C \left( 1 + \Vert x\Vert _{H^s({\mathbb {S}}^2)}^q\right) . \end{aligned}$$

(10)

Then we observe that

$$\begin{aligned} \rho u_j(t)+(1-\rho )u_j^\kappa (t) = \sum _{\ell =0}^\kappa \sum _{m=-\ell }^\ell u_j^{\ell ,m}(t)Y_{\ell , m} + \rho \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell u_1^{\ell ,m}(t)Y_{\ell , m}. \end{aligned}$$

This would imply for $\rho \in [0,1]$

$$\begin{aligned}&\Vert \varphi '(\rho u_j(t)+(1-\rho )u_j^\kappa (t))\Vert _{L^2(\varOmega ;H^s({\mathbb {S}}^2))}^2\\ &\quad \le C^2 {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \left( 1 + \Vert \rho u_j(t)+(1-\rho )u_j^\kappa (t)\Vert _{H^s({\mathbb {S}}^2)}^q \right) ^2 \right] \\ &\quad \le 2 C^2 \left( 1 + {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \left( \left\| \sum _{\ell =0}^\kappa \sum _{m=-\ell }^\ell u_j^{\ell ,m}(t)Y_{\ell , m}\right\| _{H^s({\mathbb {S}}^2)}\right. \right. \right. \\ &\qquad \left. \left. \left. + \rho \left\| \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell u_1^{\ell ,m}(t)Y_{\ell , m}\right\| _{H^s({\mathbb {S}}^2)} \right) ^{2q} \right] \right) \\ &\quad \le 2 C^2 \left( 1 + \Vert u_j(t)\Vert _{L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))}^{2q} \right) . \end{aligned}$$

Therefore Assumption 1 is satisfied if $u_j(t) \in L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))$ which is specified in Proposition 2.

Having seen that the class of test functions with derivatives of polynomial growth satisfies Assumption 1, we are in place to state our general weak convergence result.

Proposition 5

Under the setting of Proposition 4and Assumption 1, there exists a constant ${\hat{C}}$ such that the weak errors are bounded by

$$\begin{aligned}&\left| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \varphi (u_1(t)) - \varphi (u_1^\kappa (t))\right] \right| \\ &\quad \le {\hat{C}}\left( \kappa ^{-(\alpha /2+s)} +\kappa ^{-(\beta +s)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} +\kappa ^{-(\gamma +s+1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \\ &\left| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \varphi (u_2(t)) - \varphi (u_2^\kappa (t))\right] \right| \\ &\quad \le {\hat{C}}\left( \kappa ^{-(\alpha /2+s-1)} +\kappa ^{-(\beta +s-1)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} +\kappa ^{-(\gamma +s)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) \end{aligned}$$

for all $\kappa > \ell _0.$

Proof

The proof is inspired by [1]. Consider the Gelfand triple

$$\begin{aligned} V\subset H\subset V^* \end{aligned}$$

with $V=H^s({\mathbb {S}}^2), H=L^2({\mathbb {S}}^2)$ and $V^*=H^{-s}({\mathbb {S}}^2).$ The mean value theorem for Fréchet derivatives followed by the Cauchy–Schwarz inequality yields

$$\begin{aligned}&\left| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ \varphi (u_j(t))-\varphi (u_j^\kappa (t)) \right] \right| \\ &\quad = \left| {{\,\mathrm{{\mathbb {E}}}\,}}\left[ {_{V}}\langle \int _0^1\varphi '(\rho u_j(t)+(1-\rho )u_j^\kappa (t))\,\text{ d }\rho , u_j(t)-u_j^\kappa (t)\rangle _{V^*} \right] \right| \\ &\quad \le \left\| \int _0^1\varphi '(\rho u_j(t)+(1-\rho )u_j^\kappa (t))\,\text{ d }\rho \right\| _{L^2(\varOmega ;V)} \Vert u_j(t)-u_j^\kappa (t)\Vert _{L^2(\varOmega ;V^*)} \end{aligned}$$

for $j=1,2.$ The first term is bounded by Assumption 1 so that the convergence rate will be obtained from the second term. Details are only given for the first component, i.e., for $j=1,$ and are obtained for $u_2$ in a similar way.

Following the proof of Proposition 3, we obtain

$$\begin{aligned}&\left\| u_1(t)-u_1^\kappa (t)\right\| _{L^2(\varOmega ;H^{-s}({\mathbb {S}}^2))}\\ &\quad \le \Vert v_1 - v_1^\kappa \Vert _{H^{-s}({\mathbb {S}}^2)} + \left( \sum _{\ell =\kappa +1}^\infty \sum _{m=-\ell }^\ell (1+\ell (\ell +1))^{-s} (\ell (\ell +1))^{-1}|v_2^{\ell ,m}|^2\right) ^{1/2}\\ &\qquad + \Vert {\hat{Z}}-{\hat{Z}}^\kappa \Vert _{L^2(\varOmega ;L^2({\mathbb {S}}^2))} \end{aligned}$$

with

$$\begin{aligned} {\hat{Z}} = (\text{Id}-\varDelta _{{\mathbb {S}}^2})^{-s/2} Z \end{aligned}$$

and ${\hat{Z}}^\kappa$ its approximation. Therefore the angular power spectrum of the centered Gaussian random field ${\hat{Z}}$ is given by

$$\begin{aligned} {\hat{A}}_\ell = (1 + \ell (\ell +1))^{-s} {\widetilde{A}}_\ell \le C \ell ^{-(\alpha + 2 + 2s)}. \end{aligned}$$

Applying Theorem 1 to ${\hat{Z}}$ and bounding the initial conditions as in Proposition 3 with the additional weights $(1+ \ell (\ell +1))^{-s/2}$ yield

$$\begin{aligned}&\left\| u_1(t)-u_1^\kappa (t)\right\| _{L^2(\varOmega ;H^{-s}({\mathbb {S}}^2))}\\ &\quad \le C\left( \kappa ^{-(\alpha /2+s)} +\kappa ^{-(\beta +s)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^2)} +\kappa ^{-(\gamma +s+1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^2)} \right) , \end{aligned}$$

which concludes the proof for the weak error in the first component. $\square$

Proposition 4 states that the approximation of the second moment converges with twice the strong rate of convergence obtained in Proposition 3. Let us now investigate which regularity (for the noise and initial values) is required to achieve twice the strong rate in Proposition 5. We focus on the convergence of the noise in the parameter $\alpha .$ Similar considerations hold for the initial conditions.

For having the weak rate in Proposition 5 to be twice the strong rate from Proposition 3, one would need $s = \alpha /2$ for $u_1$ and $s = \alpha /2-1$ for $u_2.$

The regularity result from Proposition 2 reads $u_1(t) \in L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))$ for all $s < \alpha /2$ and $u_2(t) \in L^{2q}(\varOmega ;H^s({\mathbb {S}}^2))$ for all $s < \alpha /2-1.$ This together with the polynomial growth assumption (10) on the test functions would imply that Assumption 1 is satisfied for all $s < \alpha /2$ for $u_1$ and $s < \alpha /2-1$ for $u_2.$ Therefore, in the situation of Proposition 5, the general rule of thumb for the rate of weak convergence is also valid.

We end this section by observing that several strategies for proving weak rates of convergence of numerical solutions to SPDEs, also with multiplicative noise, in the literature could be extended to the present setting or in the case of numerical discretizations of nonlinear stochastic wave equations on the sphere, see for instance [5, 11, 14, 15, 18, 20, 23, 26, 27, 38] and references therein. This could be subject of future research.

5 Numerical experiments

We present several numerical experiments with the aim of supporting and illustrating the above theoretical results.

In order to illustrate the rate of convergence of the mean-square error from Proposition 3, we consider a reference solution at time $T=1$ with $\kappa =2^7$ (since for larger $\kappa$ the elements of the angular power spectrum $A_\ell$ and therefore the increments were so small that MATLAB failed to calculate the series expansion). The initial values are taken to be $v_1=v_2=0$ in order to observe the convergence rate only with respect to the regularity of the noise given by the parameter $\alpha .$ Afterwards we will perform a numerical example illustrating the convergence rate with respect to the regularity of the initial position given by the parameter $\beta .$ We then compute one time step of numerical solutions (since we have shown in Proposition 3 that the convergence rate is independent of the number of calculated time steps) and compute the errors for the truncation indices $\kappa =2,2^2,\ldots ,2^6.$ Instead of the $L^2({\mathbb {S}}^2)$ error in space, we used the maximum over all grid points (used for the graphical representation on the sphere) which is a stronger error. The results and the theoretical convergence rates are shown for $\alpha =3$ and $\alpha =5$ in Fig. 1, resp. Fig. 2. In all numerical experiments, we checked that the number of samples used was enough for the Monte Carlo error to be negligible.

Fig. 1

Sample and mean-square errors of the approximation of the stochastic wave equation with angular power spectrum of the Q-Wiener process with parameter $\alpha =3$ and 100 Monte Carlo samples

In these figures, one observes that the simulation results match the theoretical results from Proposition 3. In addition, in order to illustrate the structure of the solution u in dependence of the decay of the angular power spectrum, we include samples next to the convergence plots.

Fig. 2

Sample and mean-square errors of the approximation of the stochastic wave equation with angular power spectrum of the Q-Wiener process with parameter $\alpha =5$ and 100 Monte Carlo samples

In order to illustrate Remark 2 on the possibility of taking the parameter $0<\alpha <2$ to show convergence in the first component, we repeat the previous numerical experiments with $\alpha =1.$ The results are presented in Fig. 3. There, for such non-smooth noise, one can observe convergence in the position but not in the velocity. Similar observations were made for time discretizations of stochastic wave equations on domains (that are not manifolds) in [3, 10], for instance.

Fig. 3

Sample and mean-square errors of the approximation of the stochastic wave equation with angular power spectrum of the Q-Wiener process with parameter $\alpha =1$ and 100 Monte Carlo samples

In Fig. 4 we illustrate the convergence rates with respect to the regularity of the initial position from Proposition 3. To ensure that the regularity of the initial position dominates the error, we choose $\alpha =10$ and randomly an initial position $v_1$ scaled such that it belongs to $H^\beta ({\mathbb {S}}^2)$ with $\beta =2.$ The expected convergence rates are indeed observed in this figure.

Fig. 4

Mean-square errors of the approximation of the stochastic wave equation with angular power spectrum of the Q-Wiener process with parameter $\alpha =10$ and $v_1 \in H^\beta ({\mathbb {S}}^2)$ for $\beta = 2$ and 100 Monte Carlo samples

Errors of one path of the stochastic wave equation to the corresponding error plots from the previous figures (Figs. 1 and 2) are presented in Fig. 5. The observed convergence rates coincide with the theoretical results on ${\mathbb {P}}$-almost sure convergence in the second part of Proposition 3.

Fig. 5

Error of the approximation of a path of the stochastic wave equation with different angular power spectra of the Q-Wiener process

Let us now illustrate the weak rates of convergence from Proposition 4 and Proposition 5. We consider a “reference” solution at time $T=1$ with $\kappa =2^7.$ The initial values are taken to be $v_1=v_2=0.$ The test functions are given by $\varphi (u)=\Vert u\Vert _{L^2({\mathbb {S}}^2)}^2$ and $\varphi (u)=\exp (-\Vert u\Vert _{L^2({\mathbb {S}}^2)}^2).$ Observe that the second test function is of class $C^2,$ bounded and with bounded derivatives. Propositions 4 and 5 guarantee that the weak rates will be essentially twice the strong rates in both cases. This is confirmed for $\alpha =3$ in Fig. 6.

Fig. 6

Weak errors of the approximation of the stochastic wave equation with angular power spectrum of the Q-Wiener process with parameter $\alpha =3$ and 1000 Monte Carlo samples. Left column shows position, right column velocity

6 Further extensions

In this section, we extend some of the above results first to the case of the stochastic wave equation on higher-dimensional spheres ${\mathbb {S}}^{d-1},$ for some integer $d>3,$ and second to the case of a free stochastic Schrödinger equation on the sphere ${\mathbb {S}}^2.$ We keep this section concise and focus on strong and ${\mathbb {P}}$-a.s. convergence.

6.1 The stochastic wave equation on ${\mathbb {S}}^{d-1}$

Let us consider the more general situation of the stochastic wave equation on the unit sphere ${\mathbb {S}}^{d-1} = \{x \in {\mathbb {R}}^d, \Vert x\Vert _{{\mathbb {R}}^d} = 1\}$ embedded into ${\mathbb {R}}^d.$ The angular distance of two points x and y on ${\mathbb {S}}^{d-1}$ is given in the same way as on ${\mathbb {S}}^2,$ see Sect. 2. Let us denote by $(S_{\ell , m}, \ell \in {\mathbb {N}}_0, m=1,\ldots ,h(\ell ,d))$ the spherical harmonics on ${\mathbb {S}}^{d-1},$ where

$$\begin{aligned} h(\ell ,d) = (2\ell + d - 2) \frac{(\ell +d-3)!}{(d-2)!\,\ell !}. \end{aligned}$$

Using the same setup as in [32] which goes back to [39], a centered isotropic Gaussian random field Z on ${\mathbb {S}}^{d-1}$ admits a Karhunen–Loève expansion

$$\begin{aligned} Z(x) = \sum _{\ell =0}^\infty \sum _{m=1}^{h(\ell ,d)} a_{\ell , m} S_{\ell , m}(x), \end{aligned}$$

where $(a_{\ell , m}, \ell \in {\mathbb {N}}_0,m=1,\ldots ,h(\ell ,d))$ is a sequence of independent Gaussian random variables satisfying

$$\begin{aligned} {{\,\mathrm{{\mathbb {E}}}\,}}[a_{\ell , m}] = 0, \quad {{\,\mathrm{{\mathbb {E}}}\,}}[a_{\ell , m} a_{\ell ', m'}] = A_\ell \delta _{\ell \ell '} \delta _{m m'} \end{aligned}$$

for $\ell , \ell ' \in {\mathbb {N}}_0$ and $m=1,\ldots ,h(\ell ,d),$ $m' = 1,\ldots , h(\ell ',d)$ and

$$\begin{aligned} \sum _{\ell =0}^\infty A_\ell \, h(\ell ,d) < + \infty . \end{aligned}$$

The series converges with probability one and in $L^p(\varOmega ;{\mathbb {R}})$ as well as in $L^2(\varOmega ;L^p({\mathbb {S}}^{d-1})),$ $p \ge 1.$ Denoting by $(A_\ell , \ell \in {\mathbb {N}}_0)$ the angular power spectrum of Z for ${\mathbb {S}}^{d-1}$ in analogy to what was done for ${\mathbb {S}}^2,$ we can rewrite

$$\begin{aligned} Z = \sum _{\ell =0}^\infty \sum _{m=1}^{h(\ell ,d)} a_{\ell , m} S_{\ell , m} = \sum _{\ell =0}^\infty \sqrt{A_\ell } \sum _{m=1}^{h(\ell ,d)} X_{\ell , m} S_{\ell , m}, \end{aligned}$$

where $(X_{\ell , m}, \ell \in {\mathbb {N}}_0, m=1,\ldots ,h(\ell ,d))$ is the sequence of independent, standard normally distributed random variables derived by $X_{\ell , m} = a_{\ell , m}/\sqrt{A_\ell }.$ We set

$$\begin{aligned} Z^\kappa = \sum _{\ell =0}^\kappa \sqrt{A_\ell } \sum _{m=1}^{h(\ell ,d)} X_{\ell , m} S_{\ell , m} \end{aligned}$$

for the corresponding sequence of truncated random fields $(Z^\kappa ,\kappa \in {\mathbb {N}}).$ It is shown in Theorem 5.5 in [32] that these approximations converge to the random field Z in $L^p(\varOmega ;L^2({\mathbb {S}}^{d-1}))$ and ${\mathbb {P}}$-almost surely with error bounds

$$\begin{aligned} \Vert Z - Z^\kappa \Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^{d-1}))} \le C_p \cdot \kappa ^{-(\alpha +1-d)/2} \end{aligned}$$

(11)

for $\kappa > \ell _0$ and for all $\delta < (\alpha +1-d)/2$

$$\begin{aligned} \Vert Z - Z^\kappa \Vert _{L^2({\mathbb {S}}^{d-1})} \le \kappa ^{-\delta }, \quad {\mathbb {P}}\text{-a.s.}, \end{aligned}$$

(12)

where $A_\ell \le C \cdot \ell ^{-\alpha }$ for $\ell \ge \ell _0.$ This generalizes Theorem 1 above and leads to convergence rates that depend also on the dimension of the sphere.

Similarly to (2) in Sect. 2, we introduce a Q-Wiener process $(W(t), t \in {\mathbb {T}})$ on some finite interval ${\mathbb {T}}= [0,T]$ with values in $L^2({\mathbb {S}}^{d-1})$ by the expansion

$$\begin{aligned} W(t,y) = \sum _{\ell =0}^\infty \sum _{m=1}^{h(\ell ,d)} a^{\ell , m}(t) S_{\ell , m}(y) = \sum _{\ell =0}^\infty \sqrt{A_\ell } \sum _{m=1}^{h(\ell ,d)} \beta ^{\ell , m}(t) S_{\ell , m}(y), \end{aligned}$$

(13)

where $(\beta ^{\ell , m}, \ell \in {\mathbb {N}}_0, m=1,\ldots , h(\ell ,d))$ is a sequence of independent, real-valued Brownian motions.

We next recall that the Laplace–Beltrami operator $\varDelta _{{\mathbb {S}}^{d-1}}$ on ${\mathbb {S}}^{d-1}$ has the spherical harmonics $(S_{\ell , m}, \ell \in {\mathbb {N}}_0, m=1,\ldots ,h(\ell ,d))$ as eigenbasis with eigenvalues given by

$$\begin{aligned} \varDelta _{{\mathbb {S}}^{d-1}} S_{\ell , m} = - \ell (\ell +d-2)S_{\ell , m} \end{aligned}$$

for $\ell \in {\mathbb {N}}_0$ and $m=1,\ldots ,h(\ell ,d)$ (see, e.g., [4, Sec. 3.3]).

We introduce Sobolev spaces on ${\mathbb {S}}^{d-1},$ similarly to ${\mathbb {S}}^2,$ which are given for a smoothness index $s\in {\mathbb {R}}$ by

$$\begin{aligned} H^s({\mathbb {S}}^{d-1})=(\text{Id}-\varDelta _{{\mathbb {S}}^{d-1}})^{-s/2}L^2({\mathbb {S}}^{d-1}) \end{aligned}$$

together with the norm

$$\begin{aligned} \Vert f\Vert _{H^s({\mathbb {S}}^{d-1})}=\Vert (\text{Id}-\varDelta _{{\mathbb {S}}^{d-1}})^{s/2}f\Vert _{L^2({\mathbb {S}}^{d-1})} \end{aligned}$$

for some $f\in H^s({\mathbb {S}}^{d-1}).$ We also denote $H^0({\mathbb {S}}^{d-1}) = L^2({\mathbb {S}}^{d-1}).$

The stochastic wave equation on ${\mathbb {S}}^{d-1}$ is defined as

$$\begin{aligned} \partial _{tt}u(t)-\varDelta _{{\mathbb {S}}^{d-1}}u(t)={\dot{W}}(t), \end{aligned}$$

(14)

with initial conditions $u(0)=v_1\in L^2(\varOmega ;L^2({\mathbb {S}}^{d-1}))$ and $\partial _{t}u(0)=v_2\in L^2(\varOmega ;L^2({\mathbb {S}}^{d-1})),$ where $t \in {\mathbb {T}}= [0,T],$ $T < + \infty .$ The notation ${\dot{W}}$ stands for the formal derivative of the Q-Wiener process.

Denoting as before the velocity of the solution by $u_2 = \partial _{t} u_1 = \partial _{t} u,$ one can rewrite (14) as

$$\begin{aligned} \mathop {}\!\mathrm {d}X(t)&=AX(t)\,\mathop {}\!\mathrm {d}t+G\,\mathop {}\!\mathrm {d}W(t) \\ X(0)&=X_0, \end{aligned}$$

(15)

where

$$\begin{aligned} A=\left( \begin{array}{cc}0 &{} I \\ \varDelta _{{\mathbb {S}}^{d-1}} &{} 0 \end{array}\right) , \quad G=\left( \begin{array}{c}0\\ I \end{array}\right) , \quad X=\left( \begin{array}{c} u_1\\ u_2\end{array}\right) , \quad X_0=\left( \begin{array}{c} v_1\\ v_2\end{array}\right) . \end{aligned}$$

Existence of a unique mild solution follows as before.

Using the same ansatz as in Sect. 3 with respect to the spherical harmonics on ${\mathbb {S}}^{d-1}$

$$\begin{aligned} u_1(t)=\sum _{\ell =0}^\infty \sum _{m=1}^{h(\ell ,d)} u_1^{\ell ,m}(t)S_{\ell , m}\quad \text{and}\quad u_2(t)=\sum _{\ell =0}^\infty \sum _{m=1}^{h(\ell ,d)} u_2^{\ell ,m}(t)S_{\ell , m} \end{aligned}$$

(16)

we obtain

$$\begin{aligned} u_1^{\ell ,m}(t)&=v_1^{\ell ,m}+\int _0^t u_2^{\ell ,m}(s)\,\mathop {}\!\mathrm {d}s\\ u_2^{\ell ,m}(t)&=v_2^{\ell ,m}-\ell (\ell +d-2)\int _0^t u_1^{\ell ,m}(s)\,\mathop {}\!\mathrm {d}s+a^{\ell ,m}(t), \end{aligned}$$

where $v_1^{\ell ,m},$ $v_2^{\ell ,m},$ resp. $a^{\ell ,m}$ are the coefficients of the expansions of the initial values $v_1$ and $v_2,$ resp. weighted Brownian motion in the expansion of the noise (13).

Similarly to (7), the variation of constants formula yields

$$\begin{aligned} {\left\{ \begin{array}{ll} u_1^{\ell ,m}(t) &{}= R^\ell _2(t)v_1^{\ell ,m} +R^\ell _1(t)v_2^{\ell ,m} +{{\hat{W}}}_1^{\ell ,m}(t)\\ u_2^{\ell ,m}(t) &{}=-(\ell (\ell +d-2))R^\ell _1(t) v_1^{\ell ,m} +R^\ell _2(t)v_2^{\ell ,m} +{{\hat{W}}}_2^{\ell ,m}(t), \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} {{\hat{W}}}^{\ell ,m}(t) = \left( \begin{array}{c} {{\hat{W}}}_1^{\ell ,m}(t)\\ {{\hat{W}}}_2^{\ell ,m}(t) \end{array}\right) =\int _0^t R^\ell (t-s) \,\mathop {}\!\mathrm {d}a^{\ell ,m}(s) \end{aligned}$$

with

$$\begin{aligned} R^\ell (t) = \left( \begin{array}{c} R^\ell _1(t) \\ R^\ell _2(t) \end{array}\right) = \left( \begin{array}{c} (\ell (\ell +d-2))^{-1/2}\sin (t(\ell (\ell +d-2))^{1/2})\\ \cos (t(\ell (\ell +d-2))^{1/2}) \end{array}\right) \end{aligned}$$

for $\ell \ne 0$ and

$$\begin{aligned} {{\hat{W}}}^{0,0}(t) = \left( \begin{array}{c} {{\hat{W}}}_1^{0,0}(t)\\ {{\hat{W}}}_2^{0,0}(t) \end{array}\right) = \left( \begin{array}{c} \int _0^t a^{0,0}(s) \, \mathop {}\!\mathrm {d}s\\ a^{0,0}(t) \end{array}\right) . \end{aligned}$$

Note that the only change compared to Sect. 3 is the value of the coefficients given by the eigenvalues of $\varDelta _{{\mathbb {S}}^{d-1}}$ and the renaming of the spherical harmonics.

As in Sect. 4, we approximate the solution to the stochastic wave equation (15) by truncation of the series expansion at some finite index $\kappa > 0$ and obtain

$$\begin{aligned} u_1^\kappa (t_j)=\sum _{\ell =0}^\kappa \sum _{m=1}^{h(\ell ,d)} u_1^{\ell ,m}(t_j)S_{\ell , m}\quad \text{and}\quad u_2^\kappa (t_j)=\sum _{\ell =0}^\kappa \sum _{m=1}^{h(\ell ,d)} u_2^{\ell ,m}(t_j)S_{\ell , m}. \end{aligned}$$

(17)

Then replacing the eigenvalues $-\ell (\ell +1)$ with $-\ell (\ell +d-2),$ the multiplicity of the eigenvalues $2\ell + 1$ with $h(\ell ,d)$ and applying (11) and (12) instead of Theorem 1 in the proof of Proposition 3 yields directly the following extension of Proposition 3.

Proposition 6

Let $t \in {\mathbb {T}}$ and $0=t_0< \cdots < t_n = t$ be a discrete time partition for $n \in {\mathbb {N}},$ which yields a recursive representation of the solution $X=(u_1,u_2)$ of the stochastic wave equation (15) on ${\mathbb {S}}^{d-1}$ given by (16). Assume that the initial values satisfy $v_1\in H^\beta ({\mathbb {S}}^{d-1})$ and $v_2\in H^\gamma ({\mathbb {S}}^{d-1}).$ Furthermore, assume that there exist $\ell _0 \in {\mathbb {N}},$ $\alpha > 2,$ and a constant $C>0$ such that the angular power spectrum of the driving noise $(A_\ell , \ell \in {\mathbb {N}}_0)$ satisfies $A_\ell \le C \cdot \ell ^{-\alpha }$ for all $\ell > \ell _0.$ Then, the error of the approximate solution $X^\kappa =(u_1^\kappa ,u_2^\kappa ),$ given by (17), is bounded uniformly on any finite time interval and independently of the time discretization by

$$\begin{aligned}&\Vert u_1(t) - u_1^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^{d-1}))}\\ &\quad \le {\hat{C}}_p \cdot \left( \kappa ^{-(\alpha +3-d)/2}+\kappa ^{-\beta }\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^{d-1})} + \kappa ^{-(\gamma +1)}\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^{d-1})} \right) \\ &\Vert u_2(t) - u_2^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^{d-1}))}\\ &\quad \le {\hat{C}}_p \cdot \left( \kappa ^{-(\alpha +1-d)/2}+ \kappa ^{-(\beta -1)}\Vert v_1\Vert _{H^\beta ({\mathbb {S}}^{d-1})} + \kappa ^{-\gamma }\Vert v_2\Vert _{H^\gamma ({\mathbb {S}}^{d-1})} \right) \end{aligned}$$

for all $p\ge 1$ and $\kappa > \ell _0,$ where ${\hat{C}}_p$ is a constant that may depend on p,  C,  T,  and $\alpha .$

Additionally, the error is bounded uniformly in time, independently of the time discretization, and asymptotically in $\kappa$ by

$$\begin{aligned} \Vert u_1(t) - u_1^\kappa (t)\Vert _{L^2({\mathbb {S}}^{d-1})}&\le \kappa ^{-\delta }, \quad {\mathbb {P}}\text{-a.s.} \\ \Vert u_2(t) - u_2^\kappa (t)\Vert _{L^2({\mathbb {S}}^{d-1})}&\le \kappa ^{-(\delta -1)}, \quad {\mathbb {P}}\text{-a.s.} \end{aligned}$$

for all $\delta < \min ((\alpha +3-d)/2,\beta ,\gamma +1).$

6.2 The free stochastic Schrödinger equation on ${\mathbb {S}}^2$

We consider efficient simulations of paths of solutions to the free stochastic Schrödinger equation on the sphere ${\mathbb {S}}^2$

$$\begin{aligned} \mathrm {i}\partial _tu(t)=\varDelta _{{\mathbb {S}}^2}u(t)+\dot{W}(t), \end{aligned}$$

(18)

with initial condition (possibly complex-valued) $u(0)\in L^2(\varOmega ;L^2({\mathbb {S}}^2)).$ Here, the unknown $u(t)=u_R(t)+\mathrm {i}u_I(t),$ with $t\in [0,T]$ for some $T<+\infty ,$ is a complex valued stochastic process. Furthermore, the notation $\dot{W}$ stands for the formal derivative of the (real-valued) Q-Wiener process with series expansion (2).

Considering the real and imaginary parts of the above SPDE, one can rewrite (18) as

$$\begin{aligned} \begin{aligned} \mathop {}\!\mathrm {d}X(t)&=AX(t)\,\mathop {}\!\mathrm {d}t+G\,\mathop {}\!\mathrm {d}W(t)\\ X(0)&=X_0, \end{aligned} \end{aligned}$$

(19)

where

$$\begin{aligned} A=\left( \begin{array}{cc}0 &{} \varDelta _{{\mathbb {S}}^2} \\ -\varDelta _{{\mathbb {S}}^2} &{} 0 \end{array}\right) , \quad G=\left( \begin{array}{c}0\mathrm{i}\end{array}\right) , \quad X=\left( \begin{array}{c} u_R\\ u_I\end{array}\right) , \quad X_0=\left( \begin{array}{c} u_R(0)\\ u_I(0)\end{array}\right) . \end{aligned}$$

The existence of a mild form of the abstract formulation (19) of the stochastic Schrödinger equation on the sphere follows like for the above stochastic wave equation. The mild form reads

$$\begin{aligned} X(t)=\mathrm {e}^{tA}X_0+\int _0^t\mathrm {e}^{(t-s)A}G\,\mathop {}\!\mathrm {d}W(s) \end{aligned}$$

(20)

with the semigroup

$$\begin{aligned} \mathrm {e}^{tA}= \left( \begin{array}{cc} \cos (t\varDelta _{{\mathbb {S}}^2}) &{} \sin (t\varDelta _{{\mathbb {S}}^2}) \\ -\sin (t\varDelta _{{\mathbb {S}}^2}) &{} \cos (t\varDelta _{{\mathbb {S}}^2}) \end{array}\right) . \end{aligned}$$

Finally, one obtains the integral formulation of the above problem as

$$\begin{aligned} {\left\{ \begin{array}{ll} u_R(t)&{}=u_R(0)+\int _0^t\varDelta _{{\mathbb {S}}^2}u_I(s)\,\mathop {}\!\mathrm {d}s\\ u_I(t)&{}=u_I(0)-\int _0^t\varDelta _{{\mathbb {S}}^2}u_R(s)\,\mathop {}\!\mathrm {d}s-W(t). \end{array}\right. } \end{aligned}$$

As it was done for the stochastic wave equation in Sect. 3, one can make the following ansatz for the real and imaginary part of solutions to (20)

$$\begin{aligned} u_R(t)=\sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell u_R^{\ell ,m}Y_{\ell ,m}\quad \text{and}\quad u_I(t)=\sum _{\ell =0}^\infty \sum _{m=-\ell }^\ell u_I^{\ell ,m}Y_{\ell ,m} \end{aligned}$$

and find the following system of equations defining the coefficients of these expansions:

$$\begin{aligned} {\left\{ \begin{array}{ll} u_R^{\ell ,m}(t)&{}=\cos (t(\ell (\ell +1))^{1/2})v_R^{\ell ,m}+\sin (t(\ell (\ell +1))^{1/2})v_I^{\ell ,m}+{{\hat{W}}}_R^{\ell ,m}(t)\\ u_I^{\ell ,m}(t)&{}=-\sin (t(\ell (\ell +1))^{1/2})v_R^{\ell ,m}+\cos (t(\ell (\ell +1))^{1/2})v_I^{\ell ,m}+{{\hat{W}}}_I^{\ell ,m}(t), \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} {{\hat{W}}}^{\ell ,m}(t) = \left( \begin{array}{c} {{\hat{W}}}_R^{\ell ,m}(t)\\ {{\hat{W}}}_I^{\ell ,m}(t) \end{array}\right) = \left( \begin{array}{c} \int _0^t \sin ((t-s)(\ell (\ell +1))^{1/2}) \,\mathop {}\!\mathrm {d}a^{\ell ,m}(s)\\ \int _0^t \cos ((t-s)(\ell (\ell +1))^{1/2}) \,\mathop {}\!\mathrm {d}a^{\ell ,m}(s) \end{array}\right) \end{aligned}$$

and $v_R^{\ell ,m},$ resp. $v_I^{\ell ,m},$ are the coefficients of the real, resp. imaginary, part of the initial value u(0).

It is clear that the analysis from Sect. 4 can directly be extend to the case of the stochastic Schrödinger equation on the sphere (18). The errors in the truncation procedure, denoted by $u_R^\kappa$ and $u_I^\kappa ,$ of the above ansatz are given by the following proposition (presented for zero initial data for simplicity).

Proposition 7

Let $t \in {\mathbb {T}}=[0,T]$ and $0=t_0< \cdots < t_n = t$ be a discrete time partition for $n \in {\mathbb {N}},$ which yields a recursive representation of the solution $X=(u_R,u_I)$ of the stochastic Schrödinger equation on the sphere (19) with initial data $u(0)=0.$ Assume that there exist $\ell _0 \in {\mathbb {N}},$ $\alpha > 2,$ and a constant $C>0$ such that the angular power spectrum of the driving noise $(A_\ell , \ell \in {\mathbb {N}}_0)$ decays with $A_\ell \le C \cdot \ell ^{-\alpha }$ for all $\ell > \ell _0.$ Then, the error of the approximate solution $X^\kappa =(u_R^\kappa ,u_I^\kappa )$ is bounded uniformly in time and independently of the time discretization by

$$\begin{aligned} \Vert u_R(t) - u_R^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}&\le {\hat{C}}_p \cdot \kappa ^{-(\alpha /2-1)}\\ \Vert u_I(t) - u_I^\kappa (t)\Vert _{L^p(\varOmega ;L^2({\mathbb {S}}^2))}&\le {\hat{C}}_p \cdot \kappa ^{-(\alpha /2-1)} \end{aligned}$$

for all $p\ge 1$ and $\kappa > \ell _0,$ where ${\hat{C}}_p$ is a constant that may depend on p,  C,  T,  and $\alpha .$

On top of that, the error is bounded uniformly in time, independently of the time discretization, and asymptotically in $\kappa$ by

$$\begin{aligned} \Vert u_R(t) - u_R^\kappa (t)\Vert _{L^2({\mathbb {S}}^2)}&\le \kappa ^{-(\delta -1)}, \quad {\mathbb {P}}\text{-a.s.} \\ \Vert u_I(t) - u_I^\kappa (t)\Vert _{L^2({\mathbb {S}}^2)}&\le \kappa ^{-(\delta -1)}, \quad {\mathbb {P}}\text{-a.s.} \end{aligned}$$

for all $\delta < \alpha /2.$

Observe that the above rates of convergence differ from the ones in Proposition 3 for the stochastic wave equation, since the Schrödinger semigroup is unitary. The proof of this proposition follows the lines of the proof of Proposition 3. We omit it and instead present some numerical experiments illustrating these theoretical results.

We compute the errors when approximating solutions to (18) for various truncation indices $\kappa$ for a Q-Wiener process with parameter $\alpha =4.$ All other parameters are the same as in Sect. 5. In Fig. 7, we display a sample at time $T=1$ and a strong convergence plot of the real part of the numerical approximation, as well as errors in the approximation of a path (imaginary part) to the stochastic Schrödinger equation on the sphere. These illustrations are in agreement with the results from Proposition 7.

Fig. 7

Sample, mean-square errors, and error of one path of the approximation of the stochastic Schrödinger equation with angular power spectrum of the Q-Wiener process with parameter $\alpha =4$ and 100 Monte Carlo samples (for the mean-square errors)