An Adaptive Wavelet Space-Time SUPG Method for Hyperbolic Conservation Laws

被引:9
作者
Minbashian, Hadi [1 ]
Adibi, Hojatolah [1 ]
Dehghan, Mehdi [1 ]
机构
[1] Amirkabir Univ Technol, Fac Math & Comp Sci, Dept Appl Math, 424 Hafez Ave, Tehran 15914, Iran
关键词
adaptive wavelet method; hyperbolic conservation laws; spectral viscosity; streamline upwind Petrove-Galerkin (SUPG); continuous Galerkin; discontinuous Galerkin; postprocessing; MULTIRESOLUTION VISCOSITY METHOD; DISCONTINUOUS GALERKIN SCHEMES; SHALLOW-WATER EQUATIONS; FINITE-ELEMENT METHODS; SYSTEMS; CONVERGENCE;
D O I
10.1002/num.22180
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
This article concerns with incorporating wavelet bases into existing streamline upwind Petrov-Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space-time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusion. Furthermore, we add the hierarchical diffusion only in the vicinity of discontinuities using the feature of wavelet bases in detection of location of discontinuities. Also, we again use the last feature of the wavelet bases to perform a postprocessing using a denosing technique based on a minimization formulation to reduce Gibbs oscillations near discontinuities while keeping other regions intact. Finally, we show the performance of the proposed combination through some numerical examples including Burgers', transport, and wave equations as well as systems of shallow water equations. (c) 2017 Wiley Periodicals, Inc.
引用
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页码:2062 / 2089
页数:28
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