对流扩散方程的多尺度解耦小波算法研究

A multi-scale decoupling wavelet algorithm for convection-diffusion equations

针对传统有限元方法在求解对流扩散问题时常会出现的数值震荡和数值耗散等缺点,提出一种对流扩散方程的尺度解耦小波求解方法。介绍第二代小波多分辨分析,推导有限元多分辨空间的两尺度关系,提出对流扩散方程的多尺度计算框架。推导对流扩散方程的解耦条件,并利用提升方案构造多尺度解耦小波。提出多尺度解耦小波算法,该方法通过向求解域添加解耦小波,逐步逼近问题精确解。数值算例证明,解耦小波是一种求解对流扩散方程性能优良的小波基。

In order to solve the convection-diffusion problems of numerical oscillation and dissipation in traditional finite ele- ment method, a scaled decoupled wavelet solution method is proposed. Firstly, the multi-resolution analysis of second genera- tion wavelets is introduced, and the finite element two multi-resolution spacial scaling relation is derived, then the multi-scale computational framework is presented for solving convection-diffusion equations. The decoupling conditions of convection-diffu- sion equations are developed, and the multi-scale decoupling wavelets are constructed by the lifting scheme. A multi-level de- coupled wavelets algorithm is proposed for approximating the exact solution by adding the decoupled wavelets into the solving domain. Numerical example shows that the decoupled wavelets have good computational performance in solving convection-dif- fusion equations.