Optimal implicit strong stability preserving Runge-Kutta methods

Strong stability preserving (SSP) time discretizations were developed for use with spatial discretizations of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified timestep restriction. We turn to implicit Runge-Kutta methods to alleviate this timestep restriction, and present implicit SSP Runge-Kutta methods which are optimal in the sense that they preserve convex boundedness properties under the largest timestep possible among all methods with a given number of stages and order of accuracy. We consider methods up to order six (the maximal order of SSP Runge-Kutta methods) and up to eleven stages. The numerically optimal methods found are all diagonally implicit, leading us to conjecture that optimal implicit SSP Runge-Kutta methods are diagonally implicit. These methods allow a larger SSP timestep, compared to explicit methods of the same order and number of stages. Numerical tests verify the order and the SSP property of the methods.