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Multilevel local time-stepping methods of Runge-Kutta-type for wave equations. (English) Zbl 1373.65069

Summary: Local mesh refinement significantly influences the performance of explicit time-stepping methods for numerical wave propagation. Local time-stepping (LTS) methods improve the efficiency by using smaller time steps precisely where the smallest mesh elements are located, thus permitting a larger time step in the coarser regions of the mesh without violating the stability condition. However, when the mesh contains nested patches of refinement, any local time step will be unnecessarily small in some regions. To allow for an appropriate time step at each level of mesh refinement, multilevel local time-stepping (MLTS) methods have been proposed. Starting from the Runge-Kutta-based LTS methods derived by M. J. Grote et al. [SIAM J. Sci. Comput. 37, No. 2, A747–A775 (2015; Zbl 1320.65140)], we propose explicit MLTS methods of arbitrarily high accuracy. Numerical experiments with finite difference and continuous finite element spatial discretizations illustrate the usefulness of the novel MLTS methods and show that they retain the high accuracy and stability of the underlying Runge-Kutta methods.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35L05 Wave equation
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 1320.65140
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