Explicit strong stability preserving second derivative multistep methods for the Euler and Navier-Stokes equations

In this paper, we develop the explicit second derivative multistep methods (SDMMs) for the Euler and Navier-Stokes equations. The SDMMs use the calculated results from the previous steps, which can improve computational efficiency. The third-order and fourth-order SDMMs temporal discretization methods are constructed based on the order conditions. Additionally, the strong stability preserving (SSP) condition for the SDMMs is proposed, which provides the corresponding parameters for optimal SSP SDMMs. For numerical experiments, the SDMMs can achieve the order of accuracy as designed on the smooth regions, and the optimal SSP coefficient for the SDMMs is shown to be roughly the same in theory as in the numerical results. Moreover, the SDMMs have approximately twice the computational efficiency compared with the third-order and fourth-order two-derivative Runge-Kutta methods, and the SDMMs are also more efficient than the classical three-stage third-order SSP Runge-Kutta method.