一种抑制激波计算中数值振荡现象的双重小波收缩方法

A Dual Wavelet Shrinkage Procedure for Suppressing Numerical Oscillation in Shock Wave Calculation

在激波数值计算中,容易出现数值振荡的问题,振荡激烈时会掩盖真实解,为此提出了许多高精度复杂计算格式或采用人工粘性抑制数值振荡.从信号处理的角度,提出双重小波收缩方法,它能自适应提取激波数值振荡解中的真实物理解.先用局部微分求积法求解浅水波方程和理想流体Euler运动方程中的激波问题,发现其数值振荡现象严重,然后采用双重小波收缩方法对其处理,获得了无数值振荡解,它能准确捕捉激波的位置并且保持激波结构.相比于复杂的Riemann(黎曼)求解格式,借助小波收缩方法,可以采用相对简单的计算格式如微分求积法求解激波问题.

In the numerical calculation of shock waves, numerical oscillation often occurred and contaminated the real solution in serious cases. For the purpose of suppressing the numeri- cal oscillation, various complicated numerical schemes or artificial viscosity methods had been applied. From the view of signal processing, a dual wavelet shrinkage procedure was formula- ted to extract the real solution hidden in the numerical solution with oscillation. The localized differential quadrature （LDQ） method was firstly used to solve the shock wave problems gov- erned by the shallow water equations and Euler equations for ideal fluid flow, and heavy oscil- lation emerged in these cases, then the dual wavelet shrinkage procedure was employed to sup- plement the LDQ method and the results without numerical oscillation were obtained, in which not only the position of shock/rarefaction wave was captured but the shock wave structure well kept. Compared with the previous complicated schemes, the present procedure enables some relatively simple scheme such as the LDQ method to effectively solve the shock wave problems.