Numerical simulations with the Galerkin least squares finite element method for the Burgers' equation on the real line

In this work we present an efficient Galerkin least squares finite element scheme to simulate the Burgers’ equation on the whole real line and subjected to initial conditions with compact support. The numerical simulations are performed by considering a sequence of auxiliary spatially dimensionless Dirichlet’s problems parameterized by its numerical support ˜K . Gaining advantage from the well-known convective-diffusive effects of the Burgers’ equation, computations start by choosing ˜K so it contains the support of the initial condition and, as solution diffuses out, ˜K is increased appropriately. By direct comparisons between numerical and analytic solutions and its asymptotic behavior, we conclude that the proposed scheme is accurate even for large times, and it can be applied to numerically investigate properties of this and similar equations on unbounded domains

李群在数值求解偏微分方程中的新应用

随着计算机技术的提高及数值计算方法的不断完善，数值计算逐渐成为研究海洋工程领域内水波动力学的有效手段，以前难以处理的非线性现象研究课题在数值手段的帮助下也出现了求解的可能。本文所研究的李群理论在非线性偏微分方程数值求解中的应用，便是这些手段中的一种。本文的选题背景为纯粹数学理论的应用与研究，通过对非线性力学方程求解方法研究现状的分析，结合李群在偏微分方程中的应用理论，提出一种偏微分方程的降维方法：从无穷小变换出发，构建保持偏微分方程不变性的李群，导出偏微分方程的降维系统，结合非经典方法中的解曲面条件，避开群不变量解的直接求解，同时实现偏微分方程的降维简化，得出原偏微分方程的数值描述。在此基础上，本文在应用实例中通过与某些偏微分方程已知精确解的比较，结果验证了该方法的合理性及有效性。在利用该方法求解几个典型的非线性水波动力学偏微分方程时，得出了一系列关于这些方程的新的数值解，并由此揭示了一些有意义的水波物理现象。 本文研究工作的意义在于丰富了偏微分方程的数值解法，为利用数值方法处理非线性问题提供了新思路，通过对水波动力学方程的求解，为海洋潮流、波浪能的基础研究做出了必要的理论铺垫，对海洋工程的实际应用也具有一定的参考价值