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Moving mesh method for problems with blow-up on unbounded domains

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Abstract

This paper studies the numerical solutions of semilinear parabolic partial differential equations (PDEs) on unbounded spatial domains whose solutions blow up in finite time. There are two major difficulties usually in numerical solutions: the singularity of blow-up and the unboundedness. We propose local absorbing boundary conditions (LABCs) on the selected artificial boundaries by using the idea of unified approach (Brunner et al., SIAM J Sci Comput 31:4478–4496, (2010). Since the uniform fixed spatial meshes may be inefficient, we adopt moving mesh partial differential equation (MMPDE) method to adapt the spatial mesh as the singularity develops. Combining LABCs and MMPDE, we can effectively capture the qualitative behavior of the blow-up singularities in the unbounded domain. Moreover, the implementation of the combination consists of two independent parts. Numerical examples also illustrate the efficiency and the accuracy of the new method.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310028, China

    Hua Qiang & Kewei Liang

Authors
  1. Hua Qiang
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  2. Kewei Liang
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Correspondence to Kewei Liang.

Additional information

This work was partially supported by a grant of key program from the National Natural Science Foundation of China (No. 10731060, 10801120), Chinese Universities Scientific Fund (No. 2010QNA3019), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, SEM (No. J20080396) and Zhejiang Provincial Natural Science Foundation of China under Grant No. Y6110252.

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Qiang, H., Liang, K. Moving mesh method for problems with blow-up on unbounded domains. Numer Algor 59, 63–77 (2012). https://doi.org/10.1007/s11075-011-9476-3

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  • DOI: https://doi.org/10.1007/s11075-011-9476-3

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