Multi Domain WENO Finite Di erence Method with Interpolation at Sub-domain Interfaces

ABSTRACT High order nite di erence WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with nite volume WENO methods of the same order of accuracy. However a main restriction is that conservative nite di erence methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this diÆculty, in this paper we develop a multi-domain high order WENO nite di erence method which uses an interpolation procedure at the sub-domain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the e ectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and eÆcient Lagrange interpolation suÆces for the sub-domain interface treatment in the multi-domainWENO nite di erence method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the nite volume version of the same method for the same accuracy.