Recently, Buchmüller and Helzel proposed a modified dimension-by-dimension finite-volume (FV) WENO method on Cartesian grids for multidimensional nonlinear conservation laws which can retain the full order of accuracy of the underlying one-dimensional (1D) reconstruction. In this work, we extend this method to multidimensional convection–diffusion equations. The 1D sixth-order central reconstruction of the conserved quantity is utilized for discretizing the diffusion terms in which the diffusion coefficients may be nonlinear functions of the conserved quantity. Using high-order accurate conversions between edge-averaged values and edge center values of any sufficiently smooth quantity, high-order accurate convective and viscous numerical fluxes at cell interfaces are computed. The present modified FV method uses fourth-order accurate conversions for the diffusive fluxes. Numerical examples show that the present method achieves fourth-order accuracy for multidimensional smooth problems, and is suitable for the numerical simulation of viscous shocked flows.

中文翻译：

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笛卡尔网格上非线性对流扩散问题的高阶修正有限体积方法

最近，Buchmüller和Helzel在笛卡尔网格上针对多维非线性守恒定律提出了一种改进的逐维有限体积（FV）WENO方法，该方法可以保留底层一维（1D）重建的全部精度。在这项工作中，我们将该方法扩展到多维对流扩散方程。利用守恒量的一维六阶中心重构来离散化扩散项，其中扩散系数可以是守恒量的非线性函数。使用任何足够平滑量的边缘平均值和边缘中心值之间的高阶准确转换，可以计算单元界面处的高阶精确对流和粘性数值通量。本改进的FV方法对扩散通量使用四阶精确转换。数值算例表明，该方法对多维光滑问题具有四阶精度，适用于粘性冲击流的数值模拟。