Commun. Comput. Phys., 7 (2010), pp. 1027-1048.


A Weighted Runge-Kutta Method with Weak Numerical Dispersion for Solving Wave Equations

Shan Chen 1, Dinghui Yang 1*, Xiaoying Deng 1

1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China.

Received 21 May 2009; Accepted (in revised version) 17 November 2009
Available online 6 January 2010
doi:10.4208/cicp.2009.09.088

Abstract

In this paper, we propose a weighted Runge-Kutta (WRK) method to solve the 2D acoustic and elastic wave equations. This method successfully suppresses the numerical dispersion resulted from discretizing the wave equations. In this method, the partial differential wave equation is first transformed into a system of ordinary differential equations (ODEs), then a third-order Runge-Kutta method is proposed to solve the ODEs. Like the conventional third-order RK scheme, this new method includes three stages. By introducing a weight to estimate the displacement and its gradients in every stage, we obtain a weighted RK (WRK) method. In this paper, we investigate the theoretical properties of the WRK method, including the stability criteria, numerical error, and the numerical dispersion in solving the 1D and 2D scalar wave equations. We also compare it against other methods such as the high-order compact or so-called Lax-Wendroff correction (LWC) and the staggered-grid schemes. To validate the efficiency and accuracy of the method, we simulate wave fields in the 2D homogeneous transversely isotropic and heterogeneous isotropic media. We conclude that the WRK method can effectively suppress numerical dispersions and source noises caused in using coarse grids and can further improve the original RK method in terms of the numerical dispersion and stability condition.

AMS subject classifications: 65M06, 65M12, 86-08, 86A15

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Key words: WRK method, seismic wavefield modeling, anisotropy, numerical dispersion.

*Corresponding author.
Email: chenshan05@mails.tsinghua.edu.cn (S. Chen), dhyang@math.tsinghua.edu.cn (D. Yang), xydeng@math.tsinghua.edu.cn (X. Deng)
 

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