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A new class of optimal high-order strong-stability-preserving time discretization methods. (English) Zbl 1020.65064

There are several ways for finding numerical solution of time dependent partial differential equations (PDEs). One of them is a method of lines based on discretization in space and then solving a set of ordinary differential equations using standard time stepping techniques or Runge-Kutta methods. Standard stability analysis for the solver of such a systems generally focuses on linear stability and is often adequate when the desired solutions are smooth. Solutions to hyperbolic PDEs may not be smooth; shock waves or discontinuous behaviour could occur even for smooth initial data. Numerical methods mentioned above exhibit a weak form of instability (nonlinear instability) resulting in unphysical behaviour. So, numerical methods based on a nonlinear stability requirement are very useful. Originally they were reffered as total variation diminishing methods, in this and recent articles they are reffered as strong-stability-preserving (SSP) methods.
Optimal SSP schemes based on Runge-Kutta methods have been found for accuracy orders 1, 2, and 3, where the number of stages \( s\) is assumed to be equal to the order \(p\). In this paper the authors derive a new class of optimal high-order SSP Runge-Kutta schemes where the restriction \( s=p\) is lifted. Optimal high order SSP and low storage SSP Runge-Kutta schemes with \( s>p\) are investigated.
The performance of these new schemes on a few test problems is included. Test problems possess features that pose particular difficulties to numerical methods such as contact discontinuities, compressive shocks and sonic points. The results from these investigations indicate that both the standard and low-storage versions of the presented schemes confirm efficiency and good stability of the presented method in comparison to currently available methods.

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
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L70 Second-order nonlinear hyperbolic equations
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