Two barriers on strong-stability-preserving time discretization methods (Q1610529)

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 \(4\).