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

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.     

Accelerating technique for implicit finite difference schemes for two dimensional parabolic partial differential equations

In this paper we define a new accurate fast implicit method for the finite difference solution of the two dimensional parabolic partial differential equations with first level condition, which may be obtained by any other method. The stability region is discussed. The suggested method is considered as an accelerating technique for the implicit finite difference scheme, which is used to find the first level condition. The obtained results are compared with some famous finite difference schemes and it is in satisfactory agreement with the exact solution.