Totally Discrete Explicit and Semi-implicit Euler Methods for a Blow-up Problem in Several Space Dimensions

The equation ut=Δu+up with homogeneous Dirichlet boundary conditions has solutions with blow-up if p>1. An adaptive time-step procedure is given to reproduce the asymptotic behavior of the solutions in the numerical approximations. We prove that the numerical methods reproduce the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.

[1]  Hermann Brunner,et al.  Blowup in diffusion equations: a survey , 1998 .

[2]  J. C. López-Marcos,et al.  The Euler method in the numerical integration of reaction-diffusion problems with blow-up , 2001 .

[3]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[4]  J. J. L. Velázquez,et al.  Classification of singularities for blowing up solutions in higher dimensions , 1993 .

[5]  M. Pino,et al.  The problem of uniqueness of the limit in a semilinear heat equation , 1999 .

[6]  H. Brunner,et al.  Numerical analysis of semilinear parabolic problems with blow-up solutions. , 1994 .

[7]  J. Rossi,et al.  A symptotic behaviour for a numerical approximation of a parabolic problem with blowing up solutions , 2001 .

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

[9]  Tomoyasu Nakagawa,et al.  Blowing up of a finite difference solution tout = uxx + u2 , 1975 .

[10]  Frank Harary,et al.  Graph Theory , 2016 .

[11]  Yoshikazu Giga,et al.  Nondegeneracy of blowup for semilinear heat equations , 1989 .

[12]  Ricardo G. Durán,et al.  Numerical approximation of a parabolic problem with a nonlinear boundary condition , 1998 .

[13]  Robert D. Russell,et al.  Moving Mesh Methods for Problems with Blow-Up , 1996, SIAM J. Sci. Comput..

[14]  Ricardo G. Durán,et al.  An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up , 2002, Computing.

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

[16]  L. M. Abia,et al.  On the blow-up time convergence of semidiscretizations of reaction-diffusion equations , 1998 .

[17]  Marie-Noëlle Le Roux,et al.  Semidiscretization in time of nonlinear parabolic equations with blowup of the solution , 1994 .

[18]  J. C. López-Marcos,et al.  Blow-up for semidiscretizations of reaction-diffusion equations , 1996 .

[19]  A. P. Mikhailov,et al.  Blow-Up in Quasilinear Parabolic Equations , 1995 .

[20]  R. Russell,et al.  New Self-Similar Solutions of the Nonlinear Schrödinger Equation with Moving Mesh Computations , 1999 .

[21]  Yoshikazu Giga,et al.  Characterizing Blow-up Using Similarity Variables , 1985 .

[22]  T. Ushijima.  On the approximation of blow-up time for solutions of nonlinear parabolic equations , 2000 .

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

[24]  Blow-up vs. spurious steady solutions , 2001 .

[25]  Chris Budd,et al.  An invariant moving mesh scheme for the nonlinear diffusion equation , 1998 .

[26]  Peter R. Turner,et al.  Topics in Numerical Analysis , 1982 .

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

[28]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.