Spectral Methods
Algorithms, Analysis and Applications
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29 Result(s)
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Article
Stability and convergence of the variable-step time filtered backward Euler scheme for parabolic equations
This work is concerned with numerical analysis of the variable-step time filtered backward Euler scheme (see e.g. DeCaria in SIAM J Sci Comput 43(3):A2130–A2160, 2021) for linear parabolic equations. To this e...
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Article
Arbitrarily High Order and Fully Discrete Extrapolated RK–SAV/DG Schemes for Phase-field Gradient Flows
In this paper, we construct and analyze a fully discrete method for phase-field gradient flows, which uses extrapolated Runge–Kutta with scalar auxiliary variable (RK–SAV) method in time and discontinuous Gale...
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Article
Energy Plus Maximum Bound Preserving Runge–Kutta Methods for the Allen–Cahn Equation
It is difficult to design high order numerical schemes which could preserve both the maximum bound property (MBP) and energy dissipation law for certain phase field equations. Strong stability preserving (SSP)...
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Article
Efficient Stochastic Galerkin Methods for Maxwell’s Equations with Random Inputs
In this paper, we are concerned with the stochastic Galerkin methods for time-dependent Maxwell’s equations with random input. The generalized polynomial chaos approach is first adopted to convert the original...
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Article
Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations
We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation...
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Article
Numerical Blow-Up of Nonlinear Parabolic Integro-Differential Equations on Unbounded Domain
The efficient numerical methods of the nonlinear parabolic integro-differential PDEs on unbounded spatial domains whose solutions blow up in finite time are considered. Based on the unified approach proposed i...
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Article
Numerical Solutions for Weakly Singular Volterra Integral Equations Using Chebyshev and Legendre Pseudo-Spectral Galerkin Methods
In this paper we present and analyze Chebyshev and Legendre pseudo-spectral methods for the second kind Volterra integral equations with weakly singular kernel
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Article
High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes
It has been demonstrated that spectral deferred correction (SDC) methods can achieve arbitrary high order accuracy and possess good stability properties. There have been some recent interests in using high-ord...
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Article
Convergence Analysis of Spectral Galerkin Methods for Volterra Type Integral Equations
This work is to provide spectral and pseudo-spectral Jacobi-Galerkin approaches for the second kind Volterra integral equation. The Gauss-Legendre quadrature formula is used to approximate the integral operato...
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Article
Galerkin Methods for Stochastic Hyperbolic Problems Using Bi-Orthogonal Polynomials
This work is concerned with scalar transport equations with random transport velocity. We first give some sufficient conditions that can guarantee the solution to be in appropriate random spaces. Then a Galerk...
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Article
A Speed-Up Strategy for Finite Volume WENO Schemes for Hyperbolic Conservation Laws
In this paper, a speed-up strategy for finite volume WENO schemes is developed for solving hyperbolic conservation laws. It adopts p-adaptive like reconstruction, which automatically adjusts from fifth order W...
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Book
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Chapter
Fourier Spectral Methods for Periodic Problems
The spectral method was introduced in Orszag’s pioneer work on using Fourier series for simulating incompressible flows about four decades ago (cf. Orszag (1971)). The word “spectral” was probably originated f...
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Chapter
Introduction
Numerical methods for partial differential equations can be classified into the local and global categories. The finite-difference and finite-element methods are based on local arguments, whereas the spectral ...
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Chapter
Volterra Integral Equations
This chapter is devoted to spectral approximations of the Volterra integral equation (VIE): $$y(t)+\int_{o}^{t}R(t,\tau)y(\tau)=f(t),\,\,t\epsilon[0,...
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Chapter
Spectral Methods for Second-Order Two-Point Boundary Value Problems
We consider in this chapter spectral algorithms for solving the two-point boundary value problem. $$ -\varepsilon \rm{U}^{n}+ p(x){\rm{U}=F}, in 1:=(...
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Chapter
Higher-Order Differential Equations
High-order differential equations often arise from mathematical modeling of a variety of physical phenomena. For example, higher even-order differential equations may appear in astrophysics, structural mechani...
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Chapter
Separable Multi-Dimensional Domains
The main goals of this chapter are (a) to design efficient spectral algorithms for solving second-order elliptic equations in separable geometries; and (b) to provide a basic framework for error analysis of mu...
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Chapter
Orthogonal Polynomials and Related Approximation Results
The Fourier spectral method is only appropriate for problems with periodic boundary conditions. If a Fourier method is applied to a non-periodic problem, it inevitably induces the so-called Gibbs phenomenon, a...
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Chapter
Unbounded Domains
We study in this chapter spectral approximations by orthogonal polynomials/functions on unbounded intervals, such as Laguerre and Hermite polynomials /functions and rational functions. Considerable progress ha...
