Strong stability-preserving three-derivative Runge–Kutta methods

Abstract

In this work, we present the explicit strong stability-preserving (SSP) three-derivative Runge–Kutta (ThDRK) methods and propose the order accuracy conditions for ThDRK methods by Albrecht’s approach. Additionally, we develop the SSP theory based on the new Taylor series condition for the ThDRK methods and find its optimal SSP coefficient with the corresponding parameters. By comparing with two-derivative Runge–Kutta (TDRK) methods, Runge–Kutta (RK) methods and second derivative general linear methods (SGLMs), the theoretical and numerical results show that the ThDRK methods have the largest effective SSP coefficient for the order accuracy ($3\le p\le 5$). The numerical experiments reveal that the ThDRK methods maintain the designed order of convergence on the linear advection and Euler equation, and indicate the ThDRK methods have effective computational cost.

Appendix A Order conditions for the ThDRK methods

We alternate recursively compute the global stage errors ${\varvec{\epsilon }}$ and the derivative ${\varvec{\delta }}$, ${\varvec{\sigma }}$ of the global stage errors. The terms and conditions for each order accuracy of the ThDRK methods are described below.

First:

$$\begin{aligned} \begin{array}{|c|l|} \hline \text {Condition: } &{} \tau _{1}^{n+1}=0.~~~~ \\ \hline \end{array} \end{aligned}$$

Second:

Third:

Fourth:

Fifth:

Sixth:

Seventh: