Two Barriers on Strong-Stability-Preserving Time Discretization Methods

Strong-stability-preserving (SSP) time discretization methods are popular and effective algorithms for the simulation of hyperbolic conservation laws having discontinuous or shock-like solutions. They are (nonlinearly) stable with respect to general convex functionals including norms such as the total-variation norm and hence are often referred to as total-variation-diminishing (TVD) methods. For SSP Runge-Kutta (SSPRK) methods with positive coefficients, we present results that fundamentally restrict the achievable CFL coefficient for linear, constant-coefficient problems and the overall order of accuracy for general nonlinear problems. Specifically we show that the maximum CFL coefficient of an s-stage, order-p SSPRK method with positive coefficients is s-p+1 for linear, constant-coefficient problems. We also show that it is not possible to have an s-stage SSPRK method with positive coefficients and order p > 4 for general nonlinear problems.