Numerical Blow-up of Semilinear Parabolic PDEs on Unbounded Domains in ℝ2

We study the numerical solution of semilinear parabolic PDEs on unbounded spatial domains Ω in ℝ2 whose solutions blow up in finite time. Of particular interest are the cases where Ω=ℝ2 or Ω is a sectorial domain in ℝ2. We derive the nonlinear absorbing boundary conditions for corresponding, suitably chosen computational domains and then employ a simple adaptive time-stepping scheme to compute the solution of the resulting system of semilinear ODEs. The theoretical results are illustrated by a broad range of numerical examples.

[1]  Xiaonan Wu,et al.  Numerical Method for the Deterministic Kardar- Parisi-Zhang Equation in Unbounded Domains , 2006 .

[2]  H. Fujita.  On the blowing up of solutions fo the Cauchy problem for u_t=Δu+u^ , 1966 .

[3]  Houde Han.  The Artificial Boundary Method—Numerical Solutions of Partial Differential Equations on Unbounded Domains , 2006 .

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

[5]  Fernando Quirós,et al.  An adaptive numerical method to handle blow-up in a parabolic system , 2005, Numerische Mathematik.

[6]  Pingwen Zhang,et al.  Frontiers and prospects of contemporary applied mathematics , 2006 .

[7]  Jiwei Zhang,et al.  Artificial boundary method for two-dimensional Burgers' equation , 2008, Comput. Math. Appl..

[8]  Hong Jiang,et al.  Absorbing Boundary Conditions for the Schrödinger Equation , 1999, SIAM J. Sci. Comput..

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

[10]  Philippe Souplet,et al.  Uniform blow‐up profile and boundary behaviour for a non‐local reaction–diffusion equation with critical damping , 2004 .

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

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

[13]  Jiwei Zhang,et al.  Computational Solution of Blow-Up Problems for Semilinear Parabolic PDEs on Unbounded Domains , 2009, SIAM J. Sci. Comput..

[14]  Xiaonan Wu,et al.  Artificial boundary method for Burgers' equation using nonlinear boundary conditions , 2006 .

[15]  G. Strang.  On the Construction and Comparison of Difference Schemes , 1968 .

[16]  Philippe Souplet,et al.  Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source , 1999 .

[17]  Chunxiong Zheng,et al.  Exact nonreflecting boundary conditions for one-dimensional cubic nonlinear Schrödinger equations , 2006, J. Comput. Phys..

[18]  Xiaonan Wu,et al.  Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations: Two-dimensional case. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  Xiaonan Wu,et al.  Adaptive absorbing boundary conditions for Schrödinger-type equations: Application to nonlinear and multi-dimensional problems , 2007, J. Comput. Phys..

[20]  Houde Han,et al.  Absorbing boundary conditions for nonlinear Schrödinger equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Houde Han,et al.  EXACT ARTIFICIAL BOUNDARY CONDITIONS FOR QUASILINEAR ELLIPTIC EQUATIONS IN UNBOUNDED DOMAINS , 2008 .

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

[23]  Keng Deng,et al.  The Role of Critical Exponents in Blow-Up Theorems: The Sequel , 2000 .

[24]  Robert D. Russell,et al.  A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations , 2008, J. Comput. Phys..

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

[26]  Zhiwen Zhang,et al.  Split local absorbing conditions for one-dimensional nonlinear Klein-Gordon equation on unbounded domain , 2008, J. Comput. Phys..

[27]  Xiaonan Wu,et al.  Unified approach to split absorbing boundary conditions for nonlinear Schrödinger equations. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[28]  Chunxiong Zheng,et al.  Numerical Solution to the Sine-Gordon Equation Defined on the Whole Real Axis , 2007, SIAM J. Sci. Comput..

[29]  H. Keller,et al.  Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains , 1987 .

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

[31]  Howard A. Levine,et al.  The Role of Critical Exponents in Blowup Theorems , 1990, SIAM Rev..

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

[33]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .