求解含有高阶导数偏微分方程的局部间断Petrov-Galerkin方法（英文）

A Local Discontinuous Petrov-Galerkin Method for Partial Differential Equations with High Order Derivatives

构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等.

A local discontinuous Petrov-Galerkin method is proposed for solving three types of partial differential equations with second,third and fourth order derivatives,respectively.They are Burgers type equations,Kd V type equations and bi-harmonic type equations.The method extends discontinuous Petrov-Galerkin method for conservation laws by rewriting corresponding equations into a first order system and solving the system instead of the original equation.The method has a fourth order accuracy and maintains advantages of discontinuous Petrov-Galerkin method.Numerical simulations verify that the method reaches optimal convergence order and simulates well complex wave interaction such as soliton propagation and collision.