Domain Decomposition Method for fee Boundary Problems and Its Applications

已知一维双曲方程和反应扩散方程存在行波解。作为一种捕捉行波解的数值方法，我们提出了一种任意精度跟踪水平集的数值方法。该方法的特点是将水平集视为自由边界，将原问题转化为自由边界问题。自由边界问题是定义在边界未知的域上的边值问题，必须确定其解。许多实际问题被表述为自由边界问题。近年来，自由边界问题的数值方法得到了发展和改进。但由于区域的形状未知，数值结果的可靠性研究并不容易。因此，我们将区域分解法和固定域法结合起来，开发了无限精度数值模拟（IPNS）。它由谱配点法和多精度算法组成。采用谱配点法控制截断误差。采用多精度算法控制舍入误差。该方法适用于具有光滑解的PDE系统。它便于研究数值结果的可靠性。自由边界问题用IPNS求解。数值结果令人满意

It is known that there are travelling wave solutions in one-dimensional hyperbolic equations and reaction-diffusion equations. As a numerical method to capture the travelling wave solutions, we propose a numerical method for tracking the level set with the arbitrary precision. The feature of this method is that the level set is considered to be a free boundary and the original problem is transformed into a free boundary problem. Free boundary problems are boundary value problems defined on domains whose boundaries are unknown and must be determined as the solution. Many practical problems are formulated as free boundary problems. Recently, numerical methods for free boundary problems have been developed and improved. But investigation of the reliability of numerical results is not easy because of the unknown shape of the domain. So, we use the domain decomposition method and the fixed domain method togetherOtherwise, IPNS(Infinite-Precision Numerical Simulation) was developed. It consists of the spectral collocation method and multiple precision arithmetic. The spectral collocation method is used for the control of truncation errors. Multiple precision arithmetic is used for the control of rounding errors. The method is applicable to PDE systems with smooth solutions. It facilitates investigation of the reliability of numerical resultsIt is Solved by IPNS for the free boundary problem. Numerical results are very satisfactory