Numerical procedures for a boundary value problem with a non-linear boundary condition

摘要

Numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial-boundary value problem for the two-dimensional parabolic partial differential equation (PDE) with a non-linear boundary condition. This class of parabolic PDEs plays a very important role in many branches of science and engineering. The non-linear condition is in the form of a double integral giving the specification of mass in the solution domain. Not only the problem has both Neumann and Dirichlet boundary conditions but the Dirichlet boundary condition is in a non-standard form. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Due to the structure of the boundary conditions a reduced one-dimensional parabolic equation for the new unknown v(y,t)=∫10u(x,y,t)dx can be formulated. The resulting problem has a non-local boundary condition. The new two-dimensional parabolic PDE with Neumann’s boundary conditions will be solved numerically by using the method of lines semi-discretization approach. The space derivatives in the PDE are approximated by finite difference replacements. The solution of the resulting system of first-order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time is controlled by choosing several subdiagonal Pade approximants to replace this matrix exponential term. Numerical techniques are developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The article concludes with the results of some numerical experiments.