Implicit Solution of a Two-Dimensional Parabolic Inverse Problem with Temperature Overspecification

Three different implicit finite difference schemes for solving the two-dimensional parabolic inverse problem with temperature overspecification are considered. These schemes are developed for indentifying the control parameter which produces, at any given time, a desired temperature distribution at a given point in the spatial domain. The numerical methods discussed, are based on the second-order (5,1) Backward Time Centered Space (BTCS) implicit formula, and the second-order (5,5) Crank-Nicolson implicit finite difference formula and the fourth-order (9,9) implicit scheme. These finite difference schemes are unconditionally stable. The (9,9) implicit formula takes a huge amount of CPU time, but its fourth-order accuracy is significant. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) times needed for each of the methods are discussed and compared. The implicit finite difference schemes use more central processor times than the explicit finite difference techniques, but they are stable for every diffusion number.