Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation.
翻译:对于守恒律,构建了高阶有限差分加权基本非振荡(WENO)方法, 对于多维问题,它们在与相同精度的有限体积WENO或DG方案相比具有较低的成本,因此对于许多科学和工程应用具有吸引力。但据我们所知,尚未将此类方案扩展到具有非守恒乘积的非线性双曲系统。在本文中, 我们执行这种扩展,从而改进了此类方案的应用范围。扩展是通过以涟漪形式编写方案来实现的。我们使用Dumbser和Balsara (2016)的HLLI Riemann Solver作为扩展的构建块。由于使用了HLL构建块,因此所得到的方案具有适当的超音速极限。使用抗扩散通量可确保方案保存静止不连续性,从而扩展了其应用领域。我们的新有限差分WENO公式使用与经典版本相同的WENO重构,因此用户过渡到当前公式非常容易。