Summation-by-Parts Operators for General Function Spaces
Copyright policy )Summation-by-Parts Operators for General Function Spaces
Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.
22 pages, 6 figures
- Linköping University Sweden
- Dartmouth College United States
- Johannes Gutenberg University of Mainz Germany
- University of Johannesburg South Africa
Numerical optimization and variational techniques, mimetic discretization, Numerical methods for trigonometric approximation and interpolation, trigonometric functions, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, 65M12, 65M60, 65M70, 65D25, 65T40, 65D12, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, general function spaces, exponential functions, Mathematics - Numerical Analysis, summation-by-parts operators, Matematik, Numerical radial basis function approximation, radial basis functions, Numerical Analysis (math.NA), Numerical differentiation, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Spectral, collocation and related methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, Mathematics, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Numerical optimization and variational techniques, mimetic discretization, Numerical methods for trigonometric approximation and interpolation, trigonometric functions, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, 65M12, 65M60, 65M70, 65D25, 65T40, 65D12, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, general function spaces, exponential functions, Mathematics - Numerical Analysis, summation-by-parts operators, Matematik, Numerical radial basis function approximation, radial basis functions, Numerical Analysis (math.NA), Numerical differentiation, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Spectral, collocation and related methods for boundary value problems involving PDEs, Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs, Mathematics, Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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