利用小波求解二维非线性双曲型守恒律

Using Wavelet to Solve Nonlinear Hyperbolic Conservation Laws

在 C.W.Shu和 S.Osher提出的基于点值的高阶精确基本无振荡格式的基础上 ,使用插值小波变换对函数进行多尺度分解 ,直接利用点值插值误差来探测具有奇异性的区域范围 (比如包括激波的区域 )。在这些奇性区域 ,利用代价昂贵的高阶基本无振荡格式计算单元边界的数值通量 ;而在解光滑的区域 ,则用廉价的多项式插值根据先前低分辨尺度上得到的值来计算出其数值散度 ,从而减少计算工作量。一些其它的技巧也被用来提高其计算效率。最后这种方法被应用到求解二维非线性双曲型守恒律 ,同时还可以很容易地通过在一维基础上计算数值散度的方式扩展到高维情形。

Based on the high order accurate essentially non oscillatory (ENO) scheme dealing with point values that C.W. Shu and S. Osher put forword,this paper utilizes interpolating wavelet transform for giving the multi level decomposition of a function,and uses directly the interpolating errors of point values to detect regions with singularites (e.g.shocks). In these regions, an expensive high order accurate ENO scheme is applied to evaluating the numerical flux at cell boundaries. And in smooth regions a cheap polynomical interpolation is used to get the value of the numerical divergence from values previously obtained on lower resolution scales. Some other techniques are also used in this method for improving its efficiency. Finally, this method is applied to 2 D nonlinear hyperbolic conservation laws, and it can be extended from one dimension to higher dimensions easily for the computation of numerical divergence.