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

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

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Article ID: iaor201113495
Volume: 59
Issue: 1
Start Page Number: 63
End Page Number: 77
Publication Date: Jan 2012
Journal: Numerical Algorithms
Authors: ,
Keywords: differential equations
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., 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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