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Local oscillations in finite difference solutions of hyperbolic conservation laws

Authors: Jiequan Li, Huazhong Tang, Gerald Warnecke and Lumei Zhang
Journal: Math. Comp. 78 (2009), 1997-2018
MSC (2000): Primary 65M06, 65T50; Secondary 35L40, 35L65, 76M20
Published electronically: January 28, 2009
MathSciNet review: 2521276
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Abstract: It was generally expected that monotone schemes are oscillation-free for hyperbolic conservation laws. However, recently local oscillations were observed and usually understood to be caused by relative phase errors. In order to further explain this, we first investigate the discretization of initial data that trigger the chequerboard mode, the highest frequency mode. Then we proceed to use the discrete Fourier analysis and the modified equation analysis to distinguish the dissipative and dispersive effects of numerical schemes for low frequency and high frequency modes, respectively. It is shown that the relative phase error is of order $ {\mathcal O}(1)$ for the high frequency modes $ u_j^n=\lambda^n_k e^{i\xi j}$, $ \xi\approx \pi$, but of order $ {\mathcal O}(\xi^2)$ for low frequency modes ( $ \xi \approx 0$). In order to avoid numerical oscillations, the relative phase errors should be offset by numerical dissipation of at least the same order. Numerical damping, i.e. the zero order term in the corresponding modified equation, is important to dissipate the oscillations caused by the relative phase errors of high frequency modes. This is in contrast to the role of numerical viscosity, the second order term, which is the lowest order term usually present to suppress the relative phase errors of low frequency modes.

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Additional Information

Jiequan Li
Affiliation: School of Mathematics, Capital Normal University, Beijing 100037, People’s Republic of China

Huazhong Tang
Affiliation: LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China

Gerald Warnecke
Affiliation: Institut für Analysis und Numerik, Otto-von-Guericke-Universität, PSF 4120, 39016 Magdeburg, F.R. Germany

Lumei Zhang
Affiliation: The high school attached to the Central University for Nationalities, Beijing 100081, People’s Republic of China

Keywords: Finite difference schemes, high and low frequency modes, oscillations, chequerboard modes, numerical damping, numerical viscosity, relative phase error, modified equation analysis, discrete Fourier analysis.
Received by editor(s): March 14, 2008
Received by editor(s) in revised form: September 13, 2008
Published electronically: January 28, 2009
Article copyright: © Copyright 2009 American Mathematical Society