A novel Lax–Wendroff type procedure of two-derivative time-stepping schemes for Euler and Navier–Stokes equations

In this paper, we developed a novel Lax–Wendroff (LW) type procedure of two-derivative time-stepping schemes for the Euler and Navier–Stokes equations. The explicit two-derivative time-stepping schemes, including the two-derivative Runge–Kutta schemes and variable step size two-derivative multistep schemes, are proposed and the optimized time-stepping schemes are determined by the strong stability preserving theory. The novel LW procedure greatly simplifies the intricate symbolic calculations of the original LW procedure, particularly for the Navier–Stokes equations. The novel LW procedure also ensures both the order accuracy and numerical stability compared to the original LW procedure. Moreover, we presented a simplified positivity-preserving limiter for LW procedure, enabling the temporal schemes to handle demanding computations. When employing high-order spatial methods, numerical tests highlighted that two-derivative time-stepping schemes utilizing the novel LW procedure effectively improve the stability and computational efficiency compared to the Runge–Kutta schemes.