求解一维对流扩散方程的高阶方法
High-Order Methods for Solving One-Dimensional Convection-Diffusion Equations
摘要: 本文提出了一类求解一维对流扩散方程的埃尔米特插值的加权本质无振荡格式，称为HWENO (Hermite WENO)格式。这类格式的主要优点是在光滑区域内实现高阶精度，在间断处能够保持强间断性且无振荡。本文将对流扩散方程中对流项采用HWENO格式去求解，为了保证格式的紧性和高阶精度，扩散项采用三点的埃尔米特插值去近似得到，首先将方程写成守恒的半离散形式。格式的构造中，空间项基于有限体积形式的高精度Hermite重构，时间项采用非线性稳定的Runge-Kutta方法推进。大量的数值结果验证了本文格式的有效性和稳定性。
Abstract: In this paper, a class of weighted essentially non-oscillatory (WENO) schemes of Hermite interpolation, termed HWENO (Hermite WENO) schemes, for solving one-dimension convection-diffusion equations is presented. The main advantage of the schemes is their capability to achieve high order formal accuracy in smooth regions while maintaining stable, nonoscillatory and sharp discontinuity transitions. In this paper, the convection term in the convection-diffusion equation is solved by the HWENO scheme. In order to ensure the compactness and high order accuracy of the scheme, the diffusion term is approximated by the three-point Hermite interpolation. Firstly, the equation is written into a conserved semi-discrete form. The constructed spatial term was based on the high order accuracy Hermite interpolation, finite volume formulation, and the time term was advanced by using the nonlinearly stable Runge-Kutta method. A large number of numerical results verify the validity and stability of the proposed scheme.
文章引用：刘艺明, 刘红霞. 求解一维对流扩散方程的高阶方法[J]. 应用数学进展, 2021, 10(4): 1132-1140. https://doi.org/10.12677/AAM.2021.104123

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