Moving mesh methods for problems with blow-up. (English) Zbl 0860.35050

Author’s summary: “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 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 efficiency.”
The PDEs considered in the paper are mainly of the form \[ u_t = u_{xx} + f(u)\quad\text{ with } f(u)= u^p\;(p>1)\quad\text{ or } f(u)=e^u \] and \(0<x<1\), \(t>0\).


35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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