导出Burgers方程整体离散化的边界条件
Derive boundary conditions for holistic discretisations of Burgers' equation
AI总结：
基于中心流形理论，为Burgers方程的整体离散化方法导出自然边界条件，涵盖Neumann和Dirichlet型，并生成对称带宽矩阵，适用于有限域上一般时空系统的非线性演化建模。
摘要：
我之前使用Burgers方程介绍了一种新的偏微分方程数值离散化方法。该分析基于中心流形理论，因此我们确信离散化能精确模拟所有过程及其亚网格尺度相互作用。在此，我展示了如何将物理域的边界自然地纳入Burgers方程的数值建模中。我们研究了Neumann和Dirichlet边界条件。除了模拟非线性对流外，该方法还自然地导出具有恒定带宽的对称矩阵，以对应自伴扩散算子。这里开发的技术可用于在有限域上精确模拟相当一般的时空动力系统的非线性演化。
英文摘要：
I previously used Burgers' equation to introduce a new method of numerical discretisation of \pde{}s. The analysis is based upon centre manifold theory so we are assured that the discretisation accurately models all the processes and their subgrid scale interactions. Here I show how boundaries to the physical domain may be naturally incorporated into the numerical modelling of Burgers' equation. We investigate Neumann and Dirichlet boundary conditions. As well as modelling the nonlinear advection, the method naturally derives symmetric matrices with constant bandwidth to correspond to the self-adjoint diffusion operator. The techniques developed here may be used to accurately model the nonlinear evolution of quite general spatio-temporal dynamical systems on bounded domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
