Many-stage optimal stabilized Runge-Kutta methods for hyperbolic partial differential equations. (English) Zbl 1535.65163
Explicit Runge-Kutta (RK) methods are commonly considered for the integration of hyperbolic partial differential equations (PDEs). Stable timestep needs to be significantly reduced due to the CFL condition. To increase computational efficiency, stabilized explicit RK methods have been introduced. Stabilized explicit RK methods use additional stages to improve the stability properties of the scheme, allowing larger timesteps. In this paper, an optimization approach is devised for the generation of optimal stability polynomials for spectra of hyperbolic PDEs. The optimization approach relies on the properties of the pseudo-extrema for the proven optimal stability polynomials of first and second order for disks. Optimal stability polynomials for both convex and nonconvex spectra are presented. Stability polynomials with degrees larger than 100 are constructed for a range of classical hyperbolic PDEs that match the linear consistency requirements up to order three. Numerical schemes are constructed by minimizing the propagation and amplification of round-off errors, which is challenging for many-stage methods. For linear problems, only internal stability might limit the theoretically possible maximum timestep while for nonlinear problems the lack of the strong stability preserving (SSP) property spoils the effectiveness of the very high-stage methods.
Reviewer: Bülent Karasözen (Ankara)
MSC:
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D07 | Numerical computation using splines |
| 65K10 | Numerical optimization and variational techniques |
| 65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |
| 49M41 | PDE constrained optimization (numerical aspects) |
| 41A50 | Best approximation, Chebyshev systems |
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