求解Hamilton-Jacobi方程的一类高精度差分格式

构造了一、二维非线性Hamilton-Jacobi方程的一类新的高精度高分辨率差分格式.首先将计算区域划分为互不重叠的子单元,再根据格式的精度要求分割子单元为细小于单元,其次通过子单元上各个细小子单元节点的函数值构造空间导数的高阶插值逼近,为避免由此产生的数值振荡,对空间导数在各节点左右侧的值进行TVD/TVB校正,利用高阶Runge-Kutta TVD时间离散方法得到一维Hamilton-Jacobi方程的高阶全离散格式并推广到二维情况,最后给出了几个典型的数值算例,验证了格式具有计算简单、高分辨间断导数、无振荡等特性.

A Class of High Order Accurate Difference Schemes for Hamilton-Jacobi Equations

A new class of high order accuracy, high resolution difference schemes is presented for one dimensional and two dimensional nonlinear Hamilton-Jacobi equations. Firstly, the computational domain is divided into many pieces of non-overlapping subcells, and then each subcell is further subdivided into subsubcells according to required accuracy; Secondly, by using the nodal point values in each subcell, high-order interpolating approximation are used to compute the spatial derivative. Furthermore, TVD/TVB correction are given to prevent oscillations from the high-order interpolation. Afterwards, by means of high-order Runge-Kutta TVD time discretization, a high-order fully discretization method is obtained. The extension to two dimensional Hamilton-Jacobi equations is also carried out. Finally, several classic numerical tests are given, which show that simplicity in computation, high-order accuracy, high-resolution discontinuities in the derivatives and non-oscillatory behaviors of the resulting schemes.