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修正有限差分HWENO方法求解对流扩散方程
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Abstract:
在本文中，我们提出了一种修正的高阶有限差分Hermite WENO (HWENO)方法，用于求解均匀网格中的一维和二维对流扩散方程。与求解双曲守恒律的有限差分HWENO方法不同，我们扩展了该方法以求解对流扩散方程。其关键不是使用通量分裂技术，而是使用添加高阶修正项的想法来提高数值通量的精度。此外，在重构过程中，我们不在单元界面上使用函数及其导数值，而是使用解及其导数的点值直接插值。使用Hermite插值计算高阶导数和扩散项，以保持方法的紧凑性。这种方法的一个优点是数值通量的重构过程可以采用任意单调通量。另一个优点是，修改后的方法仍然具有HWENO方案的紧性，并且在相同的网格上也具有更小的数值误差和更好的分辨率。通过一维和二维问题的数值算例验证了所提方法的有效性和稳定性。
In this paper, we propose a modified high-order finite difference Hermite WENO (HWENO) method for solving one and two dimensions convection-diffusion equations in uniform meshes. Unlike the finite difference HWENO method for solving hyperbolic conservation laws, we extend the method to solve convection-diffusion equations. The key is not to use the flux splitting technique, but to use the idea of adding higher-order corrections to improve the precision of the numerical flux. Moreover, in the reconstruction process, we do not use the function and its derivative values on the cell interface, but use the direct interpolation of the point values of the solution and its derivatives. The higher derivatives and diffusion term are computed using Hermite interpolation to maintain the compactness of the method. An advantage of this method is that the reconstruction process of the numerical flux can adopt any monotone flux. Another advantage is that the modified method still has the compactness of the HWENO schemes, and also has smaller numerical errors and better resolution on the same mesh. The validity and stability of the proposed method are verified by numerical examples of one and two dimensions problems.
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