Strong Stability Preserving Two-step Runge–Kutta Methods

Abstract

We investigate the strong stability preserving (SSP) property of two-step Runge-Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge-Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations. * 4700

Computation of optimal monotonicity preserving general linear methods

“…The present results imply that these methods, which in some cases were obtained by nonlinear optimization, are indeed optimal, even among much larger classes of methods than those considered in [12,15]. These results also imply the optimality of the thirdorder SSP general linear methods with up to three stages or three steps in [12,15].…”

“…They can be The second-order methods are of particular interest because they are also optimal second-order SSP methods in their respective classes. For instance, the optimal second-order two-stage methods have been proposed as SSP methods in [12], while the optimal second-order two-step methods have been proposed in [15]. The present results imply that these methods, which in some cases were obtained by nonlinear optimization, are indeed optimal, even among much larger classes of methods than those considered in [12,15].…”

“…For instance, the optimal second-order two-stage methods have been proposed as SSP methods in [12], while the optimal second-order two-step methods have been proposed in [15]. The present results imply that these methods, which in some cases were obtained by nonlinear optimization, are indeed optimal, even among much larger classes of methods than those considered in [12,15]. These results also imply the optimality of the thirdorder SSP general linear methods with up to three stages or three steps in [12,15].…”

Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

“…1.2) and SSP conditions (14a)-(14b) are all satisfied. Based on this, we wrote a Matlab optimization code for finding optimal twoderivative multistage methods [8], formulated along the lines of David Ketcheson's code [19] for finding optimal SSP multistage multistep methods in [2,17]. We used this to find optimal SSP multistage two-derivative methods of order up to p = 5.…”

“…Explicit multistep SSP methods of order p > 4 do exist, but have severely restricted time-step requirements [7]. Explicit multistep multistage methods that are SSP and have order p > 4 have been developed as well [2,17].…”