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A Time Second-Order Mass-Conserved Implicit-Explicit Domain Decomposition Scheme for Solving the Diffusion Equations

  • Zhongguo Zhou (a1) (a2) and Dong Liang (a3)

In the paper, a new time second-order mass-conserved implicit/explicit domain decomposition method (DDM) for the diffusion equations is proposed. In the scheme, firstly, we calculate the interface fluxes of sub-domains from the obtained solutions and fluxes at the previous time level, for which we apply high-order Taylor’s expansion and transfer the time derivatives to spatial derivatives to improve the accuracy. Secondly, the interior solutions and fluxes in sub-domains are computed by the implicit scheme and using the relations between solutions and fluxes, without any correction step. The mass conservation is proved and the convergence order of the numerical solutions is proved to be second-order in both time and space steps. The super-convergence of numerical fluxes is also proved to be second-order in both time and space steps. The scheme is stable under the stable condition r≤3/5. The important feature is that the proposed domain decomposition scheme is mass-conserved and is of second order convergence in time. Numerical experiments confirm the theoretical results.

Corresponding author
*Corresponding author. Email: (Z. G. Zhou), (D. Liang)
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[1] Aziz, K. and Settari, A., Petroleum Reservoir Simulation, Applied Science Publisher, Ltd., London, 1979.
[2] Bear, J., Hydraulics of Groundwater, McGraw-Hill, New York, 1978.
[3] Amitai, D., Averbuch, A., Israeli, M. and Itzikowitz, S., Implicit-explicit parallel asyn-chronous solver of parabolic PDEs, SIAM J. Sci. Comput., 19 (1998), pp. 13661404.
[4] Dawson, C. N., Du, Q. and Dupont, T. F., A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math. Comput., 57 (1991), pp. 6371.
[5] Dryja, M. and Tu, X., A domain decomposition discretization of parabolic problems, Numerische Mathematik, 107 (2007), pp. 625640.
[6] Du, Q., Mu, M. and Wu, Z., Efficient parallel algorithms for parbolic problems, SIAM J. Numer. Anal., 39 (2001), pp. 14691487.
[7] Du, C. and Liang, D., An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media, J. Comput. Phys., 229 (2010), pp. 45014521.
[8] Evans, D. and Abdullah, A., Group explicit methods for parabolic equations, Int. J. Comput. Math., 14 (1983), pp. 73105.
[9] Gaiffe, S., Glowinski, R. and Lemonnier, R., Domain decomposition and splitting methods for parabolic equations via a mixed formula, in the 12th International Conference on Domain Decomposition, Chiba, Japan, 1999.
[10] Kuznetsov, Y., New algorithms for approximate realization of implicit difference scheme, Soviet J. Numer. Anal. Math. Model., 3 (1988), pp. 99114.
[11] Liang, D. and Du, C., The efficient S-DDM scheme and its analysis for solving parabolic equations, I. Comput. Phys., 272 (2014), pp. 4669.
[12] Shi, H. and Liao, H., Unconditional stability of corrected explicit-implicit domain decomposition algorithms for parallel approximation of heat equations, SIAM J. Numer. Anal., 44 (2006), pp. 15841611.
[13] Zhu, S., Conservative domain decomposition procedure with unconditional stability and second-order accuracy, Appl. Math. Comput., 216 (2010), pp. 32753282.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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