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Runge-Kutta approximation for -semigroups in the graph norm with applications to time domain boundary integral
Alexander Rieder1, Francisco-Javier Sayas2, Jens Markus Melenk3
1Fakultät für Mathematik, Universität Wien, Vienna, Austria.
SN Partial Differential Equations and Applications
|January 18, 2021
Summary
This study introduces novel approximation techniques for evolution problems using A-stable Runge-Kutta methods. New estimates in graph norms improve numerical methods for wave scattering and heat conduction.
Area of Science:
- Numerical Analysis
- Applied Mathematics
- Scientific Computing
Background:
- Abstract evolution problems are fundamental in modeling dynamic systems.
- Approximating solutions with inhomogeneous side constraints poses significant numerical challenges.
- A-stable Runge-Kutta methods are widely used for their stability properties in solving differential equations.
Purpose of the Study:
- To develop and analyze novel a priori estimates for abstract evolution problems.
- To extend these estimates to norms beyond the standard Banach space, specifically the graph norm of the generator.
- To apply these improved estimates to enhance the accuracy and stability of numerical discretizations.
Main Methods:
- Utilizing A-stable Runge-Kutta methods for the approximation of abstract evolution problems.
- Deriving novel a priori estimates in non-standard norms, including the graph norm of the generator.
- Applying convolution quadrature techniques for discretizing specific physical problems.
Main Results:
- Established new a priori estimates for the approximation of abstract evolution problems.
- Demonstrated the effectiveness of these estimates in the graph norm of the generator.
- Successfully applied the developed methods to problems in wave scattering and heat conduction, showing improved numerical properties.
Conclusions:
- The derived estimates provide a stronger theoretical foundation for numerical methods applied to evolution problems.
- The approach enhances the accuracy and reliability of discretizations for complex physical phenomena.
- This work contributes to the advancement of numerical techniques in scientific computing and applied mathematics.
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