Lax-Wendroff solvers-based Hermite reconstruction for hyperbolic problems

•The main contribution is to develop a new family of Hermite reconstructions for hyperbolic problems systematically based on Lax-Wendroff flow solvers, and implement them in the two-stage fourth order time-stepping framework.•The Hermite-type reconstruction and Lax-Wendroff type time discretization are unified to form compact spacetime coupling schemes.•The resulting scheme becomes quite compact and efficient. Numerical results demonstrate the performance of this approach. This paper develops a new family of 2k-th order accurate Lax-Wendroff solvers-based Hermite reconstructions for hyperbolic problems systematically and implements the resulting schemes in the two-stage fourth order time-stepping framework. In order to cope with numerical difficulties around discontinuities, the WENO technology is applied and in practice a hybrid choice is made between a second order (k=1) and any other higher order Hermite reconstructions (k≥2) for efficiency. This approach unifies the Hermite-type reconstructions and the Lax-Wendroff type time discretization to form compact spacetime coupling schemes. Numerical experiments demonstrate the performance of the resulting schemes.