A robust finite difference method for a singularly perturbed degenerate parabolic problem II

作者：Viscor Martin; Stynes Martin^*

来源：IMA Journal of Numerical Analysis, 2013, 33(2): 460-480.

DOI：10.1093/imanum/drs011

• singularly perturbed
• degenerate parabolic problem
• Shishkin mesh

The parabolic singularly perturbed problem epsilon u(xx)(x,t) - x(alpha) u(t)(x,t) = f(x,t) is considered on the rectangular domain = (0,1) x (0,T] with Dirichlet initial and boundary conditions. Here, epsilon is a small positive parameter and alpha is a positive constant. This problem is degenerate since the coefficient x(alpha) of u(t) vanishes along the side x = 0 of . Bounds on the derivatives of u are used to design a nonuniform mesh and a finite difference method on this mesh is constructed to solve the problem numerically. As the solution u is not in general uniformly bounded with respect to epsilon in the maximum norm, the convergence analysis of the numerical method requires the use of some unusual barrier functions and a special weighted discrete norm. Numerical examples are provided to support the theoretical results.

• 出版日期2013-4