Multidimensional flux difference splitting schemes

Anovel alteration to the Cauchy-Kowalevski procedure is here presented to obtain essentially monotonic solutions for multidimensional flows. It is argued that this can be accomplished by splitting the cross-derivative terms among the several dimensions, such that the coefficient of the cross derivatives remains small compared to the coefficient of the normal derivatives. The approach naturally lends itself to extending the Roe flux difference splitting scheme to multiple dimensions and is advantaged over previous Cauchy-Kowalevski-based methods by yielding a solution free of spurious oscillations in the vicinity of oblique shock waves. Several test cases ranging from low-speed subsonic flows in channels to hypersonic flows over ramp injectors indicate that the proposed genuinely multidimensional method generally achieves a twofold or more increase in resolution along each dimension over the dimensionally split Roe scheme while retaining its appealing attributes: the scheme has a compact three-node-bandwidth stencil, is a finite volume flux function, yields essentially monotonic solutions, introduces minimal dissipation within viscous layers, and is written in general matrix form. Although the method proposed is first-order accurate, it offers a resolution as high or higher than the dimensionally split second-order total-variation-diminishing schemes for many problems of interest and is expected to surpass significantly the latter when extended to second-order accuracy.