Abstract
In this paper, the stability analysis and optimal error estimates are presented for a kind of fully discrete schemes for solving one-dimensional linear convection–diffusion equation with periodic boundary conditions. The fully discrete schemes are defined with local discontinuous Galerkin (LDG) spatial discretization methods coupled with implicit–explicit (IMEX) temporal discretization methods based on backward difference formulas (BDF). By combining an improved multiplier technique used in the stability analysis for multistep methods and the technique to deal with derivative and jump in LDG methods, we establish a general framework of stability analysis for the corresponding fully discrete LDG–IMEX–BDF schemes up to fifth order in time. The considered schemes are proved to be unconditionally stable, in the sense that a properly defined “discrete energy” is dissipative if the time step is upper bounded by a constant which is independent of the mesh size. Optimal orders of the \(L^2\) norm accuracy in both space and time are also proved by energy analysis. Numerical tests are presented to validate the theoretical results.
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References
Adams, R.A.: Sobolev Spaces. Academic Press, Canbridge (1975)
Akrivis, G.: Stability of implicit–explicit backward difference formulas for nonlinear parabolic equations. SIAM J. Numer. Anal. 53(1), 464–484 (2015)
Akrivis, G.: Stability properties of implicit–explicit multistep methods for a class of nonlinear parabolic equations. Math. Comput. 85(301), 2217–2229 (2016)
Akrivis, G., Chen, M., Yu, F., Zhou, Z.: The energy technique for the six-step BDF method. SIAM J. Numer. Anal. 59(5), 2449–2472 (2021)
Akrivis, G., Katsoprinakis, E.: Backward difference formulae: new multipliers and stability properties for parabolic equations. Math. Comput. 85(301), 2195–2216 (2016)
Akrivis, G., Lubich, C.: Fully implicit, linearly implicit and implicit–explicit backward difference formulae for quasi-linear parabolic equations. Numer. Math. 131(4), 713–735 (2015)
Ascher, U., Ruuth, S., Spiteri, R.: Implicit–explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997)
Ascher, U., Ruuth, S., Wetton, B.: Implicit explicit methods for time-dependent partial-differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)
Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(2), 267–279 (1997)
Calvo, M., de Frutos, J., Novo, J.: Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations. Appl. Numer. Math. 37(4), 535–549 (2001)
Castillo, P., Cockburn, B., Schotzau, D., Schwab, C.: Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection–diffusion problems. Math. Comput. 71(238), 455–478 (2002)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)
Cockburn, B., Shu, C.-W.: Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)
Dahlquist, G.: G-stability is equivalent to A-stability. BIT Numer. Math. 18(4), 384–401 (1978)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics. Springer, Berlin (1996)
Hu, J., Shu, R.: On the uniform accuracy of implicit–explicit backward differentiation formulas (IMEX–BDF) for stiff hyperbolic relaxation systems and kinetic equations. Math. Comput. 90(328), 641–670 (2021)
Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225(2), 2016–2042 (2007)
Kennedy, C., Carpenter, M.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)
Koto, T.: IMEX Runge–Kutta schemes for reaction–diffusion equations. J. Comput. Appl. Math. 215(1), 182–195 (2008)
Lubich, C., Mansour, D., Venkataraman, C.: Backward difference time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal. 33(4), 1365–1385 (2013)
Nevanlinna, O., Odeh, F.: Multiplier techniques for linear multistep methods. Numer. Funct. Anal. Optim. 3(4), 377–423 (1981)
Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for advection–diffusion problems. SIAM J. Numer. Anal. 53(1), 206–227 (2015)
Wang, H., Shu, C.-W., Zhang, Q.: Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems. Appl. Math. Comput. 272(2), 237–258 (2016)
Wang, H., Wang, S., Zhang, Q., Shu, C.-W.: Local discontinuous Galerkin methods with implicit–explicit time-marching for multi-dimensional convection–diffusion problems. ESAIM: M2AN 50(4), 1083–1105 (2016)
Wang, H., Zhang, Q., Shu, C.-W.: Third order implicit–explicit Runge–Kutta local discontinuous Galerkin methods with suitable boundary treatment for convection–diffusion problems with Dirichlet boundary conditions. J. Comput. Appl. Math. 342, 164–179 (2018)
Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Comm. Comput. Phys. 7(1), 1–46 (2010)
Zhang, Q., Gao, F.: A fully-discrete local discontinuous Galerkin method for convection-dominated Sobolev equation. J. Sci. Comput. 51(1), 107–134 (2012)
Zhao, W., Huang, J.: Boundary treatment of implicit–explicit Runge–Kutta method for hyperbolic systems with source terms. J. Comput. Phys. 423, 109828 (2020)
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Hai** Wang: Research is supported by National Natural Science Foundation of China (Grant No. 12071214). Qiang Zhang: Research is supported by National Natural Science Foundation of China (Grant No. 12071214).
Appendix
Appendix
1.1 A.1: The coefficients in Lemmas 3.1 and 4.2 for \(s=1\) and \(s=2\)
For \(s=1\) and \(s=2\), the coefficients of \(\kappa _1\), \(\kappa _2\), \(\{d_i\}_{i=1}^{s-1}\) and the entries of matrix \({\mathbb {G}}\) are well known, see [16]. Specifically, for \(s=1\), we have
and for \(s=2\), we have
Obviously, \(\kappa _1>0\) and \(\kappa _2>0\). The positive-definiteness of \({\mathbb {G}}\) can be checked easily. Furthermore, in (3.2) we can simply take \(\lambda _0 = \beta \) for these two cases, thus \(A=0\) and \(\{c_i\}_{i=1}^{s}=0\).
Similarly, in (4.13) we can take \({\tilde{\lambda }}_0=\varepsilon _0=\frac{\beta }{2}\) for these two cases, so \(B=0\) and \(\{e_i\}_{i=1}^{s}=0\).
1.2 A.2: The coefficients in Lemmas 3.1 and 4.2 for \(s=3\)
For \(s=3\), \(\kappa _1\), \(\kappa _2\), \(\{d_i\}_{i=1}^2\) and the entries of matrix \({\mathbb {G}}\) are the same as that presented in [17], namely,
Obviously, \(\kappa _1>0\) and \(\kappa _2>0\). The positive-definiteness of \({\mathbb {G}}\) can be verified by checking its eigenvalues. In fact, the eigenvalues of \({\mathbb {G}}\) are approximately equal to 0.0010644088, 0.07207072500, 1.553380287.
Taking \(\lambda _0=\frac{3}{11}\), we obtain one of the solutions for \(\{c_i\}_{i=1}^3\) and the entries of matrix \({\mathbb {A}}\), they are given as
where \(z_*\) is one of the real root of \(10106041 z^2+(2536842-858330\sqrt{30})z -143856\sqrt{30} + 793476\). \(z_*= \frac{135 \sqrt{30}- 399 \pm 3\sqrt{-9725 + 4014\sqrt{30}}}{3179}\), which are approximately equal to 0.2115784868 and 0.0025929210, both values give positive \(a_{11}\), and thus the matrix \({\mathbb {A}}\) is positive-definite. Taking \(z_*\approx 0.0025929210\), we get the eigenvalues of \({\mathbb {A}}\) are approximately 0.0143040939 and 0.2727272727.
Further taking \({\tilde{\lambda }}_0=\frac{3}{11}\) and \(\varepsilon _0=\frac{1}{11}\), we obtain one of the solutions for \(\{e_i\}_{i=1}^3\) and the entries of matrix \({\mathbb {B}}\), they are given as
where \({\tilde{z}}_*\) is one of the real root of \(10106041 z^2+(3455573-858330\sqrt{30})z -143856\sqrt{30} + 793476\). \({\tilde{z}}_*= \frac{270 \sqrt{30}- 1087 \pm \sqrt{194665 - 11556\sqrt{30}}}{6358}\), which are approximately equal to 0.1186381176 and 0.00462419918, both values give positive \(b_{11}\), and thus the matrix \({\mathbb {B}}\) is positive-definite. Taking \({\tilde{z}}_*\approx 0.00462419918\), we get the eigenvalues of \({\mathbb {B}}\) are approximately 0.01633537215 and 0.1818181818.
1.3 A.3: The coefficients in Lemmas 3.1 and 4.2 for \(s=4\)
For \(s=4\), \(\kappa _1\), \(\kappa _2\), \(\{d_i\}_{i=1}^3\) and the entries of matrix \({\mathbb {G}}\) are given as follows, in the sense that (3.1) is satisfied with tolerance error of \(10^{-31}\) for each coefficient of \(w_iw_j\).
We observe that \(\kappa _1 >0\) and \(\kappa _2 >0\), and we can get that the eigenvalues of \({\mathbb {G}}\) are 0.00004826514323140203, 0.009410411961448486, 0.11980927324105947, 2.9619227015891934, thus \({\mathbb {G}}\) is positive-definite.
Taking \(\lambda _0=\frac{3}{25}\), we obtain one of the solutions for \(\{c_i\}_{i=1}^4\) and the entries of matrix \({\mathbb {A}}\), they are given as follows in the sense that (3.2) is satisfied with tolerance error of \(10^{-33}\) for each coefficient of \(w_iw_j\).
The eigenvalues of \({\mathbb {A}}\) are 0.1302686625796966, 0.18757782163771705, 0.43953132055618094, hence \({\mathbb {A}}\) is also positive-definite.
Further taking \({\tilde{\lambda }}_0=\frac{3}{25}\) and \(\varepsilon _0=\frac{1}{25}\), we obtain one of the solutions for \(\{e_i\}_{i=1}^4\) and the entries of matrix \({\mathbb {B}}\), they are given as follows in the sense that (4.13) is satisfied with tolerance error of \(10^{-33}\) for each coefficient of \(w_iw_j\).
The eigenvalues of \({\mathbb {B}}\) are 0.07674019909822653, 0.13793872805414528, 0.39050608894404326, hence \({\mathbb {B}}\) is also positive-definite.
1.4 A.4: The coefficients in Lemmas 3.1 and 4.2 for \(s=5\)
For \(s=5\), \(\kappa _1\), \(\kappa _2\), \(\{d_i\}_{i=1}^4\) and the entries of matrix \({\mathbb {G}}\) are given as follows, in the sense that (3.1) is satisfied with tolerance error of \(10^{-31}\) for each coefficient of \(w_iw_j\).
We can find that both \(\kappa _1\) and \(\kappa _2\) are positive, and we can compute the eigenvalues of \({\mathbb {G}}\), which are 0.000047412145709790024, 0.0011462051548831534, 0.011159345435408816, 0.1293966749328927, 3.0850671842272526, and thus \({\mathbb {G}}\) is positive-definite.
Taking \(\lambda _0 = \frac{30}{137}\), we obtain one of the solutions for \(\{c_i\}_{i=1}^5\) and the entries of matrix \({\mathbb {A}}\), they are given as follows in the sense that (3.2) is satisfied with tolerance error of \(10^{-33}\) for each coefficient of \(w_iw_j\).
The eigenvalues of \({\mathbb {A}}\) are approximately equal to 0.06856358136371647, 0.08113451794744742, 0.1278081222274782, 0.21897810218978103, so \({\mathbb {A}}\) is also positive-definite.
Further taking \({\tilde{\lambda }}_0 = \frac{30}{137}\) and \(\varepsilon _0 = \frac{3}{137}\), we obtain one of the solutions for \(\{e_i\}_{i=1}^5\) and the entries of matrix \({\mathbb {B}}\), they are given as follows in the sense that (4.13) is satisfied with tolerance error of \(10^{-33}\) for each coefficient of \(w_iw_j\).
The eigenvalues of \({\mathbb {B}}\) are approximately equal to 0.03471657383908016, 0.05359801159997174, 0.09241291077984705, 0.19708029197080293, so \({\mathbb {B}}\) is also positive-definite.
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Wang, H., Shi, X. & Zhang, Q. Stability and Error Estimates of Local Discontinuous Galerkin Methods with Implicit–Explicit Backward Difference Formulas up to Fifth Order for Convection–Diffusion Equation. J Sci Comput 96, 37 (2023). https://doi.org/10.1007/s10915-023-02264-9
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DOI: https://doi.org/10.1007/s10915-023-02264-9
Keywords
- Stability
- Error estimate
- Implicit–explicit
- Backward difference formulas
- Local discontinuous Galerkin method
- Convection–diffusion equation