Abstract
In the numerical solution for nonlinear hyperbolic equations, numerical oscillation often shows and hides the real solution with the progress of computation. Using wavelet analysis, a dual wavelet shrinkage procedure is proposed, which allows one to extract the real solution hidden in the numerical solution with oscillation. The dual wavelet shrinkage procedure is introduced after applying the local differential quadrature method, which is a straightforward technique to calculate the spatial derivatives. Results free from numerical oscillation can be obtained, which can not only capture the position of shock and rarefaction waves, but also keep the sharp gradient structure within the shock wave. Three model problems—a one-dimensional dam-break flow governed by shallow water equations, and the propagation of a one-dimensional and a two-dimensional shock wave controlled by the Euler equations—are used to confirm the validity of the proposed procedure.
Highlights
IntroductionMost problems governed by hyperbolic PDEs in fluid dynamics engineering have to be solved numerically
Due to the nonlinearity, most problems governed by hyperbolic PDEs in fluid dynamics engineering have to be solved numerically
We propose a dual wavelet shrinkage procedure to suppress numerical oscillation from a straightforward numerical scheme, named localized differential quadrature (LDQ) method, to calculate shock wave problem
Summary
Most problems governed by hyperbolic PDEs in fluid dynamics engineering have to be solved numerically. Efforts in this field led to the proposal of total variance diminishing (TVD) [3, 4] and weighted essentially nonoscillating (WENO) schemes [5, 6] Besides these sophisticated schemes, Shyy proposed a nonlinear filtering algorithm to eliminate numerical oscillation from second order central or upwind differencing in calculation of shock wave [7]. Wavelet analysis is characterized by decomposing the signal to be analyzed into multiscale coefficients; high frequency component is described by coefficients on small scale and low frequency component is described by coefficients on large scale [9, 10] In this way, shock wave may be maintained and the numerical oscillation around the shock wave may be removed after some special treatment for wavelet coefficients. We propose a dual wavelet shrinkage procedure to suppress numerical oscillation from a straightforward numerical scheme, named localized differential quadrature (LDQ) method, to calculate shock wave problem. The results are compared to their analytical solutions and very well agreement is achieved
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