A two-stage fourth order time-accurate discretization for Lax–Wendroff type flow solvers II. High order numerical boundary conditions

Abstract

This paper serves to treat boundary conditions numerically with high order accuracy in order to suit the two-stage fourth-order finite volume schemes for hyperbolic problems developed in J. Li and Z. Du (2016) [17]. As such, it is significant when capturing small scale structures near physical boundaries. Different from previous contributions in literature, the current approach constructs a fourth order accurate approximation to boundary conditions by only using the Jacobian matrix of the flux function (characteristic information) instead of its successive differentiation of governing equations leading to tensors of high ranks in the inverse Lax–Wendroff method. Technically, data in several ghost cells are constructed with interpolation so that the interior scheme can be implemented over boundary cells, and theoretical boundary condition has to be modified properly at intermediate stages so as to make the two-stage scheme over boundary cells fully consistent with that over interior cells. This is nonintuitive and highlights the fact that theoretical boundary conditions are only prescribed for continuous partial differential equations (PDEs), while they must be approximated in a consistent way (even though they could be exactly valued) when the PDEs are discretized. Several numerical examples are provided to illustrate the performance of the current approach when dealing with general boundary conditions.