Inverse Lax-Wendroff Boundary Treatment for Solving Conservation Laws with Finite Volume Methods

In this paper, we concentrate on high-order boundary treatments for finite volume methods solving hyperbolic conservation laws. The complex geometric physical domain is covered by a Cartesian mesh, resulting in the boundary intersecting the grids in various fashions. We propose two approaches to evaluate the cell averages on the ghost cells near the boundary. Both of them start from the inverse Lax-Wendroff (ILW) procedure, in which the normal spatial derivatives at inflow boundaries can be obtained by repeatedly using the governing equations and boundary conditions. After that, we can get an accurate evaluation of the ghost cell average by a Taylor expansion joined with high-order extrapolation, or by a Hermite extrapolation coupling with the cell averages on some "artificial" inner cells. The stability analysis is provided for both schemes, indicating that they can avoid the so-called "small-cell" problem. Moreover, the second method is more efficient under the premise of accuracy and stability. We perform numerical experiments on a collection of examples with the physical boundary not aligned with the grids and with various boundary conditions, indicating the high-order accuracy and efficiency of the proposed schemes.