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High order spatial discretizations with monotonicity properties are oftendesirable for the solution of hyperbolic PDEs. These methods can advantageouslybe coupled with high order strong stability preserving time discretizations.The search for high order strong stability time-stepping methods with largeallowable strong stability coefficient has been an active area of research overthe last two decades. This research has shown that explicit SSP Runge--Kuttamethods exist only up to fourth order. However, if we restrict ourselves tosolving only linear autonomous problems, the order conditions simplify and thisorder barrier is lifted: explicit SSP Runge--Kutta methods of any linear orderexist. These methods reduce to second order when applied to nonlinear problems.In the current work we aim to find explicit SSP Runge--Kutta methods with largeallowable time-step, that feature high linear order and simultaneously have theoptimal fourth order nonlinear order. These methods have strong stabilitycoefficients that approach those of the linear methods as the number of stagesand the linear order is increased. This work shows that when a high linearorder method is desired, it may be still be worthwhile to use methods withhigher nonlinear order.

Optimal Explicit Strong Stability Preserving Runge--Kutta Methods with High Linear Order and optimal Nonlinear Order