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A Newton multigrid method for steady-state shallow water equations with topography and dry areas

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Abstract

A Newton multigrid method is developed for one-dimensional (1D) and two-dimensional (2D) steady-state shallow water equations (SWEs) with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady-state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.

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Authors and Affiliations

  1. LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Kailiang Wu

  2. HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Huazhong Tang

  3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan Province, China

    Huazhong Tang

Authors
  1. Kailiang Wu
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  2. Huazhong Tang
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Correspondence to Huazhong Tang.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 91330205 and 11421101) and the National Key Research and Development Program of China (No. 2016YFB0200603)

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Wu, K., Tang, H. A Newton multigrid method for steady-state shallow water equations with topography and dry areas. Appl. Math. Mech.-Engl. Ed. 37, 1441–1466 (2016). https://doi.org/10.1007/s10483-016-2108-6

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  • DOI: https://doi.org/10.1007/s10483-016-2108-6

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Chinese Library Classification

2010 Mathematics Subject Classification

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