The Convergence of Spectral and Finite Difference Methods for Initial-Boundary Value Problems

The general theory of compatibility conditions for the differentiability of solutions to initial-boundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high-order...

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Veröffentlicht in:SIAM journal on scientific computing Jg. 23; H. 5; S. 1731 - 1751
Hauptverfasser: Flyer, Natasha, Swarztrauber, Paul N.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 2002
Schlagworte:
Algebra
Applied mathematics
Boundary conditions
Boundary value problems
Exact sciences and technology
Mathematics
Methods
Neighborhoods
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Wave equation
Spectral method
Boundary value problem
Initial condition
Numerical solution
Initial value problem
Compatibility
Partial differential equation
Convergence
Finite difference method
ISSN:1064-8275, 1095-7197
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Abstract The general theory of compatibility conditions for the differentiability of solutions to initial-boundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high-order methods. Explicit application of boundary conditions (BCs) that are independent of the initial condition (IC) results in the compatibility conditions not being satisfied. Since this is the case in most science and engineering applications, it is shown that not only does the error in a spectral method, as measured in the maximum norm, converge algebraically, but the accuracy of finite differences is also reduced. For the heat equation with a parabolic IC and Dirichlet BCs, we prove that the Fourier method converges quadratically in the neighborhood of t=0 and the boundaries and quartically for large t when the first-order compatibility conditions are violated. For the same problem, the Chebyshev method initially yields quartic convergence and exponential convergence for t>0. In contrast, the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t. These results naturally direct attention to finite difference methods that are also algebraically convergent. In the case of the wave equation, we prove that a second-order finite difference method is reduced to 4/3-order convergence and numerically show that a fourth-order finite difference scheme is apparently reduced to 3/2-order. Finally, for the wave equation subject to general ICs and zero BCs, we give a conjectureon the error for a second-order finite difference scheme, showing that an O(N-2log N) convergence is possible.
AbstractList The general theory of compatibility conditions for the differentiability of solutions to initial-boundary value problems is well known. This paper introduces the application of that theory to numerical solutions of partial differential equations and its ramifications on the performance of high-order methods. Explicit application of boundary conditions (BCs) that are independent of the initial condition (IC) results in the compatibility conditions not being satisfied. Since this is the case in most science and engineering applications, it is shown that not only does the error in a spectral method, as measured in the maximum norm, converge algebraically, but the accuracy of finite differences is also reduced. For the heat equation with a parabolic IC and Dirichlet BCs, we prove that the Fourier method converges quadratically in the neighborhood of t=0 and the boundaries and quartically for large t when the first-order compatibility conditions are violated. For the same problem, the Chebyshev method initially yields quartic convergence and exponential convergence for t>0. In contrast, the wave equation subject to the same conditions results in inferior convergence rates with all spectral methods yielding quadratic convergence for all t. These results naturally direct attention to finite difference methods that are also algebraically convergent. In the case of the wave equation, we prove that a second-order finite difference method is reduced to 4/3-order convergence and numerically show that a fourth-order finite difference scheme is apparently reduced to 3/2-order. Finally, for the wave equation subject to general ICs and zero BCs, we give a conjectureon the error for a second-order finite difference scheme, showing that an O(N-2log N) convergence is possible.
Author Swarztrauber, Paul N.
Flyer, Natasha
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Cites_doi 10.2307/2005811
10.1002/cpa.3160300602
10.1016/S0045-7825(98)00358-2
10.1090/S0025-5718-1978-0501995-4
10.1002/cpa.3160230210
10.1007/BF02566671
10.1016/0022-0396(82)90075-4
10.1002/cpa.3160230304
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Issue 5
Keywords Wave equation
Spectral method
Boundary value problem
Initial condition
Numerical solution
Initial value problem
Compatibility
Partial differential equation
Convergence
Finite difference method
Language English
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Snippet The general theory of compatibility conditions for the differentiability of solutions to initial-boundary value problems is well known. This paper introduces...
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SubjectTerms Algebra
Applied mathematics
Boundary conditions
Boundary value problems
Exact sciences and technology
Mathematics
Methods
Neighborhoods
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Sciences and techniques of general use
Title The Convergence of Spectral and Finite Difference Methods for Initial-Boundary Value Problems
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