Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order

摘要

When evolving in time the solution of a hyperbolic partial differentialequation, it is often desirable to use high order strong stability preserving(SSP) time discretizations. These time discretizations preserve themonotonicity properties satisfied by the spatial discretization when coupledwith the first order forward Euler, under a certain time-step restriction.While the allowable time-step depends on both the spatial and temporaldiscretizations, the contribution of the temporal discretization can beisolated by taking the ratio of the allowable time-step of the high ordermethod to the forward Euler time-step. This ratio is called the strongstability coefficient. The search for high order strong stability time-steppingmethods with high order and large allowable time-step had been an active areaof research. It is known that implicit SSP Runge-Kutta methods exist only up tosixth order. However, if we restrict ourselves to solving only linearautonomous problems, the order conditions simplify and we can find implicit SSPRunge-Kutta methods of any linear order. In the current work we aim to findvery high linear order implicit SSP Runge-Kutta methods that are optimal interms of allowable time-step. Next, we formulate an optimization problem forimplicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods withlarge linear stability regions that pair with known explicit SSP Runge-Kuttamethods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairsthat have high linear order and nonlinear orders p=2,3,4. These methods arethen tested on sample problems to verify order of convergence and todemonstrate the sharpness of the SSP coefficient and the typical behavior ofthese methods on test problems.