Moving Mesh Methods for Problems with Blow-Up

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.

[1]  I. M. Gel'fand,et al.  Some problems in the theory of quasilinear equations , 1987 .

[2]  D. Kassoy THE SUPERCRITICAL SPATIALLY HOMOGENEOUS THERMAL EXPLOSION: INITIATION TO COMPLETION , 1977 .

[3]  A. Chorin Estimates of intermittency, spectra, and blow-up in developed turbulence , 1981 .

[4]  Y. Giga,et al.  Asymptotically self‐similar blow‐up of semilinear heat equations , 1985 .

[5]  A. Friedman,et al.  Blow-up of positive solutions of semilinear heat equations , 1985 .

[6]  J. Dold ANALYSIS OF THE EARLY STAGE OF THERMAL RUNAWAY. , 1985 .

[7]  G. Papanicolaou,et al.  The Focusing Singularity of the Nonlinear Schrödinger Equation , 1987 .

[8]  E. Dorfi,et al.  Simple adaptive grids for 1-d initial value problems , 1987 .

[9]  Joke Blom,et al.  A moving grid method for one-dimensional PDEs based on the method of lines , 1988 .

[10]  R. Kohn,et al.  A rescaling algorithm for the numerical calculation of blowing-up solutions , 1988 .

[11]  J. Verwer,et al.  A numerical study of three moving-grid methods for one-dimensional partial differential equations which are based on the method of lines , 1990 .

[12]  J. Hyman,et al.  Dynamic rezone methods for partial differential equations in one space dimension , 1989 .

[13]  A. Stuart,et al.  On the computation of blow-up , 1990, European Journal of Applied Mathematics.

[14]  V. Galaktionov,et al.  Single point blow-up for N-dimensional quasilinear equations with gradient diffusion and source , 1991 .

[15]  M. Floater Blow-up at the boundary for degenerate semilinear parabolic equations , 1991 .

[16]  J. Bebernes,et al.  Final time blowup profiles for semilinear parabolic equations via center manifold theory , 1992 .

[17]  Weizhang Huang,et al.  Moving Mesh Methods Based on Moving Mesh Partial Differential Equations , 1994 .

[18]  Weizhang Huang,et al.  Moving mesh partial differential equations (MMPDES) based on the equidistribution principle , 1994 .

[19]  J. Vázquez,et al.  Blow-Up for Quasilinear Heat Equations Described by Means of Nonlinear Hamilton–Jacobi Equations , 1996 .