基于Hopf-Cole变换的Burgers方程初边值问题的Sinc-Galerkin法

The Sinc-Galerkin methods of the Burgers’equation based on the Hopf-Cole transformation

利用Sinc-Galerkin法数值求解Burgers方程的初边值问题。首先,用Hopf-Cole变换将二阶非线性的Burgers方程变换为二阶线性方程,同时把第一类边界条件变为第二类边界条件。时间上的导数采用θ加权格式离散,空间导数采用Sinc-Galerkin法离散,端点处分别引入权函数处理变换后的第二类边界条件。最后,通过数值算例验证了Sinc-Galerkin法的指数收敛性,与精确解相比,本文构造的数值格式精度高,能够有效捕捉激波等物理现象。

In this paper,the Sinc-Galerkin method is used to solve the initial boundary value problem of the Burgers equation.Firstly,the Hopf-Cole transform is used to transform the second-order nonlinear Burgers equation into a second-order linear equation,while the first type of boundary condition is changed into the second type of boundary condition.Then the time derivative is discretized in θ-weighted scheme,and the spatial derivative is discretized by the Sinc-Galerkin method.For the second type of boundary condition,the basis functions are introduced at the ends based on Hermite interpolation method.Finally,the validity and exponential convergence of the Sinc-Galerkin method is verified by numerical examples.A comparison between the numerical solution and the exact solution shows that numerical scheme constructed in this paper has high accuracy,and can effectively capture physical phenomena such as shock waves.