Numerical Blow-Up of Nonlinear Parabolic Integro-Differential Equations on Unbounded Domain

Abstract

The efficient numerical methods of the nonlinear parabolic integro-differential PDEs on unbounded spatial domains whose solutions blow up in finite time are considered. Based on the unified approach proposed in Zhang et al. (Phys Rev E 78:026709, 2008), Zhang et al. (Phys Rev E 79:046711, 2009), the nonlinear absorbing boundary conditions for one-dimensional and two-dimensional nonlinear parabolic integro-differential PDEs are derived. Thus the original problem on the unbounded spatial domain is reduced to an initial-boundary-value (IBV) problem on a bounded computational domain. Secondly, a simple but efficient adaptive time-stepping scheme for the reduced IBV problem is achieved by using the fixed point method to make the finite difference approximation stable at each time level. At each time level, we also prove that the lower bound and upper bound of the blow-up time can be bounded by the numerical blow-up times of the forward and backward Euler schemes. Finally, the theoretical results are illustrated by a broad range of numerical examples, including a problem with a circle line blow-up.