小波插值方法自适应数值求解时间进化微分方程

Numerical solution for differential evolutional equation using adaptive interpolation wavelet method

应用小波自相关函数的插值性质,得到任意给定函数的插值小波表达式,然后对其直接求导,可以得到函数导数的表达式。导数运算不再应用差分算法,扩展了小波方法在数值求解微分方程中的应用。由于小波基函数的有限支撑特点,小波方法可以有效地处理微分方程中解的局部突变问题。通过设定小波系数阀值,实现了求解过程的自适应。本文给出了两个算例,结果表明了算法的自适应特点及其向二维空间问题推广的有效性。

A wavelet interpolating expression for a given function is obtained using interpolating property of autocorrelation of wavelet function, and then we take differentiation to this expression to calculate the function＇s derivative. Thus, differentiation calculation is not operated by difference, but by wavelet ba- ses, therefore, it intensifies wavelet method＇s application in numerical solution of differential equation. With the aid of compactly wavelet bases, wavelet method can solve differential equation with local sharp transition solution effectively. The adaptation of solution process is realized by setting a threshold value for wavelet coefficients. Two examples including a two dimensional Burgers equation, were given in this paper to demonstrate effectiveness of this algorithm and its extension in two dimensional space.