Inverse Lax-Wendroff Method For Numerical Boundary Conditions Of Conservation Laws Based On Finite Volume Methods

In this paper,a high order inverse Lax-Wendroff method for numerical boundary conditions is developed in the finite volume framework to solve hyperbolic conservation laws on a Cartesian mesh.Because of the wide stencil of the interior high order scheme and the arbitrary shapes that boundary intersects grids with,it is difficult to give appropriate values on ghost cells.The inverse Lax-Wendroff scheme in the finite difference framework developed by Tan and Shu[32][33][34]can give relatively satisfying numerical boundary conditions of conservation laws.It is really hard to use this method directly in the finite volume framework especially on two-dimensional cases,although the finite volume scheme is more suitable for conservation laws compared with the finite difference scheme.In order to overcome this problem,a simplified finite volume inverse Lax-Wendroff scheme is designed in this paper,and a finite volume WENO type extrapolation is also developed near the boundary which can maintain accuracy and control oscillations compared with a Lagrange type extrapolation.At inflow boundaries,the inverse Lax-Wendroff scheme writes the zero and first degree normal derivatives in terms of the time derivatives and the tangential derivatives by given partial differential equations and boundary conditions,and the second and higher degree normal derivatives are determined by extrapolations.Put those normal derivatives into the Taylor expansion to get values on Gauss points near the boundary;then obtain values on ghost cells.At outflow boundaries,the normal derivatives needed are directly imposed by extrapolations.At last,some numerical tests are provided to illustrate the inverse Lax-Wendroff scheme can achieve high order and have good performance on discontinuities near the boundary.