New schemes for a two-dimensional inverse problem with temperature overspecification

Two different finite difference schemes for solving the two-dimensional par abolic inverse problem with temperature overspecification are considered. T hese schemes are developed for indentifying the control parameter which pro duces, at any given time, a desired temperature distribution at a given poi nt in the spatial domain. The numerical methods discussed, are based on the (3,3) alternating direction implicit (ADI) finite difference scheme and th e (3,9) alternating direction implicit formula. Thew schemes are unconditio nally stable. The basis of analysis of the finite difference equation consi dered here is the modified equivalent partial differential equation approac h, developed from the 1974 work of Warming and Hyett [17]. This allows dire ct and simple comparison of the errors associated with the equations as wel l as providing a means to develop more accurate finite difference schemes. These schemes use less central processor times than the fully implicit sche mes for two-dimensional diffusion with temperature overspecification. The a lternating direction implicit schemes developed in this report use more CPU times than the fully explicit finite difference schemes, but their uncondi tional stability is significant. The results of numerical experiments are p resented, and accuracy and the Central Processor (CPU) times needed for eac h of the methods are discussed. We also give error estimates in the maximum norm for each of these methods.