Numerical methods for two-dimensional parabolic inverse problem with energy overspecification

Abstract

An inverse problem concerning the two-dimensional diffusion equation with source control parameter is considered. Four finite-difference schemes are presented for identifying the con- trol parameter which produces, at any given time, a desired energy distribution in a portion of the spatial domain. The fully explicit schemes developed for this purpose, are based on the (1,5) forward time centred space (FTCS) explicit formula, and the (1,9) FTCS scheme, are economical to use, are second-order and have bounded range of stability. Therange of stability for the 9-point finite difference scheme is less restrictive than the (1,5) FTCS formula. The fully implicit finite difference schemes employed, are based on the (5,1) backward time centred space (BTCS) formula, and the (5,5) Crank–Nicolson implicit scheme, which are unconditionally stable, but use more CPU times than the fully explicit techniques. The basis of analysis of the finite difference equation considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyeet. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference methods. The results of numerical experiments are presented, and central processor (CPU) times needed for solving this inverse problem are reported.