In the solution of hyperbolic partial differential equations, it is often important to guarantee a strong boundedness condition, such as positivity or total variation boundedness.
Strong stability preserving (SSP) time discretization methods provide a theoretical guarantee of such nonlinear stability properties whenever they are satisfied under forward Euler integration, and as such are essential components of method-of-lines discretizations of hyperbolic PDEs.
We consider the problem of finding explicit and implicit Runge-Kutta methods with optimal SSP timestep restrictions. Our results for implicit methods lead to the disappointing conjecture that the effective SSP coefficient for such methods is at most two. The results for explicit methods are more encouraging. By using alternate formulations of the associated optimization problems and introducing a new, more general class of low-storage implementations of Runge-Kutta methods, new optimal low-storage methods and new low-storage implementations of known optimal methods are found. The results include families of low-storage second- and third-order methods that achieve the maximum theoretically achievable effective SSP coefficient (independent of stage number), as well as low-storage fourth-order methods that are more efficient than current full-storage methods. The theoretical properties of these methods are confirmed by numerical experiment.