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Two barriers on strong-stability-preserving time discretization methods. (English) Zbl 1003.65107
The authors study systems of ordinary differential equations obtained from the methods of lines applied to the hyperbolic conservation law \[ u_t+ f(u)_x= 0 \] with appropriate initial and boundary conditions. They note that the usual linear stability analysis in not effective for schemes of problems having discontinuous or shock-like solutions. In such cases strong-stability-preserving (SSP) methods are needed.
The authors discuss Runge-Kutta type methods SSPRK having positive coefficients. The main results of the paper concern the conditions which a Runge-Kutta method with positive coefficients have to fulfill to be SSP. These refer to the coefficients and the order of the method which cannot be superior to \(4\).

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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