Moving mesh methods for problems with blow-up

Chris J. Budd, Weizhang Huang, Robert D. Russell

Research output: Contribution to journalArticle

124 Citations (Scopus)

Abstract

In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs).Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a. (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as $t \to T$ (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy.
Original language English 305-327 23 SIAM Journal on Scientific Computing 17 2 https://doi.org/10.1137/S1064827594272025 Published - Mar 1996

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Moving Mesh Method
Moving Mesh
Blow-up
Partial differential equations
Partial differential equation
Invariance
Mesh
Scaling
Non-uniform Mesh
Blow-up Time
Blow-up Solution
Qualitative Behavior
Ignition
Efficacy
Monitor
Numerical Solution
kernel
Verify
Numerical Examples

Cite this

Moving mesh methods for problems with blow-up. / Budd, Chris J.; Huang, Weizhang; Russell, Robert D.

In: SIAM Journal on Scientific Computing, Vol. 17, No. 2, 03.1996, p. 305-327.

Research output: Contribution to journalArticle

Budd, Chris J. ; Huang, Weizhang ; Russell, Robert D. / Moving mesh methods for problems with blow-up. In: SIAM Journal on Scientific Computing. 1996 ; Vol. 17, No. 2. pp. 305-327.
title = "Moving mesh methods for problems with blow-up",
abstract = "In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs).Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a. (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as $t \to T$ (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy.",
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