双曲偏微分方程的局部伪弧长方法研究

摘要

In this paper ,a local pseudo arc-length method is proposed for hyperbolic partial differential e-quation with singular problem of shock waves ,and the forms of space transformation and adaptive mesh refinement are analyzed for the global pseudo arc-length method .In order to improve the computational efficiency ,the local pseudo arc-length method which gives the ways to determine the position of singular points and select the computational stencil is presented according to the properties of shock wave .The modifications of the new method involve how to introduce the arc-length parameters and how to dispose the shock wave oscillation .The feasibility of the local pseudo arc-length method in capturing and track-ing shock is proved through numerical examples ,and the superiority of local pseudo arc-length method in dealing with hyperbolic partial differential equation is shown by comparing our method with Godunov method for disposing different initial conditions of the hyperbolic problems .The numerical results demonstrate that our new method can be applied to engineering problems .%重点研究了局部伪弧长方法在处理偏微分方程，尤其是双曲型偏微分方程出现激波间断的奇异性问题，对比分析了全局伪弧长方法空间转化的形式及其网格自适应的性质。为提高求解效率，提出了局部伪弧长方法，利用激波间断的性质，给出了判断奇异点位置以及模板选择的方法，涉及如何处理激波振荡，如何引入弧长参数，以及怎样求解间断等问题。通过数值算例验证了局部伪弧长在激波捕捉和追踪方面的可行性，通过比较局部伪弧长方法与Godunov方法处理不同初值条件的双曲问题，显示出局部伪弧长方法处理双曲偏微分方程的优越性，为伪弧长方法应用到物理问题奠定基础。