基于高阶有限差分格式的Inverse Lax-Wendroff方法及其稳定性分析

发布时间：2018-01-14 04:27

本文关键词：基于高阶有限差分格式的Inverse Lax-Wendroff方法及其稳定性分析　出处：《中国科学技术大学》2017年博士论文　论文类型：学位论文

【摘要】：用高阶有限差分格式数值求解有限区域上偏微分方程(组)的初边值问题时,边界条件的处理非常重要,它直接影响数值方法的相容性、稳定性和精度。本文主要将Inverse Lax-Wendroff方法(简称ILW方法)与简化Inverse Lax-Wendroff 方法(Simplified Inverse Lax-Wendroff method,简称 SILW 方法)用于求解双曲守恒律方程和扩散方程初边值问题高阶有限差分格式的边界处理中;并运用GKS(Gustafsson,Kreiss and Sundstrom)分析和特征谱可视化方法分析格式的稳定性。首先,我们将ILW方法及SILW方法用于求解一维双曲守恒律方程(组)初边值问题高阶迎风格式的边界处理之中。边界条件处理主要有两个问题:一是,高阶有限差分方法需要较大的模板,导致在边界点附近需要定义虚拟点值;另一是,给定的物理边界不在网格格点上。这都会给构造有效的数值边界条件带来困难。ILW方法及SILW方法可以很好的解决上述两个问题并得到稳定且具有高精度的格式。ILW方法主要运用方程形式将边界点处的空间导数转化为已知边界条件的时间导数进而运用在相应边界点处的Taylor展开得到虚拟点的值。若方程形式很复杂或所求空间导数阶数较高时,ILW方法会造成复杂的代数运算。一种简化的ILW方法,即SILW方法,可以减少计算的复杂性和降低计算消耗。SILW方法仍运用在边界点处Taylor展开对虚拟点进行赋值,但边界点处空间导数值由以下两种方式得到:(1)由ILW方法得到;(2)由拉格朗日外推得到。运用上述两种方法构造了有效的数值边界条件后我们运用GKS分析和特征谱可视化方法分析格式的稳定性以获得相应参数取值,最后通过数值算例验证稳定性分析结果。此外,我们将SILW方法扩展到求解一维扩散方程初边值问题的高阶中心差分格式中。主要考虑了 Dirichlet边界和Neumann边界两种边界条件。对于双曲守恒律方程来说,边界点处的所有阶空间导数值均可以由ILW方法求出。但对于带Dirichlet边界条件的扩散方程,只有偶数阶导数可以由方程形式和边界条件求出。对于带Neumann边界条件的扩散方程,只有奇数阶导数可以由方程形式和边界条件求出。针对上述两种不同的边界条件,我们分别寻求了相应的方法求解边界处空间导数值,进而运用Taylor展开对虚拟点赋值,构造了有效的数值边界条件。而后运用GKS分析和特征谱可视化方法分析格式稳定性并得到保证格式稳定的相应参数取值,最后给出数值算例验证了算法。
[Abstract]:The boundary condition is very important in solving the initial boundary value problem of partial differential equations (systems) in finite domain by using higher order finite difference scheme, which directly affects the consistency of numerical methods. In this paper, the Inverse Lax-Wendroff method (ILW method) and the simplified Inverse Lax-Wendroff method (ILW method) are introduced. Simplified Inverse Lax-Wendroff method. SILW method is applied to the boundary treatment of high-order finite difference schemes for solving hyperbolic conservation law equations and diffusion equation initial-boundary value problems. The stability of the format is analyzed by GKS Gustafssonn Kreiss and Sundstrom and the method of feature spectrum visualization. We apply the ILW method and the SILW method to the boundary treatment of higher order upwind schemes for solving one-dimensional hyperbolic conservation law equations (systems). The higher order finite difference method requires a large template, which leads to the need to define virtual point values near the boundary point. The other is. The given physical boundary is not on the grid lattice point. This will bring difficulties to the construction of effective numerical boundary conditions. ILW method and SILW method can solve these two problems well and get stability and high precision. The scheme of degree. ILW method mainly uses the equation form to transform the space derivative at the boundary point into the time derivative of the known boundary condition and then use the Taylor expansion at the corresponding boundary point to obtain the value of the virtual point. If the form of the equation is very complex or the order of the space derivative is higher. The ILW method can result in complex algebraic operations. A simplified ILW method, the SILW method. It can reduce the computational complexity and reduce the computational cost. The SILW method is still used to assign the virtual point at the boundary point by Taylor expansion. However, the spatial conductance at the boundary point is obtained by the following two ways: 1) the ILW method; 2) derived from Lagrangian extrapolation. Using the above two methods to construct effective numerical boundary conditions, we use GKS analysis and feature spectrum visualization method to analyze the stability of the scheme to obtain the corresponding parameter values. Finally, the stability analysis results are verified by numerical examples. We extend the SILW method to the higher order central difference scheme for solving the initial boundary value problem of one-dimensional diffusion equation. Two boundary conditions, Dirichlet boundary and Neumann boundary, for hyperbolic conservation law equations. All the spatial derivative values at the boundary point can be obtained by the ILW method, but for the diffusion equations with Dirichlet boundary conditions. Only the even-order derivatives can be obtained from the form of equations and boundary conditions. For diffusion equations with Neumann boundary conditions. Only the odd-order derivatives can be obtained from the equation form and boundary conditions. For the above two different boundary conditions, we seek corresponding methods to solve the spatial derivative values at the boundary respectively. Then the virtual point is assigned by Taylor expansion. An effective numerical boundary condition is constructed, and then GKS analysis and feature spectrum visualization are used to analyze the stability of the scheme and obtain the corresponding parameter values to ensure the stability of the scheme. Finally, a numerical example is given to verify the algorithm.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.82

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