SLOWLY DIVERGENT SPACE MARCHING SCHEMES IN THE INVERSE HEAT CONDUCTION PROBLEM

Abstract

Recently developed “slowly divergent” space marching difference schemes, coupled with Tikhonov regularization, can solve the one-dimensional inverse heal conduction problem at values of the nondimensional lime step δt⁺ as low as δt⁺ = 0.0003. A Lax-Richtmyer analysis is used to demonstrate dramatic differences in error amplification behavior among various space marching algorithms, for the same problem, on the same mesh; maximum error amplification factors may differ by more than 10 orders of magnitude at parameter values that are of interest in rocket nozzle applications. Slowly divergent schemes are characterized by their damping behavior at high frequencies. A widely used benchmark problem, where the surface temperature gradient is a step function, provides a basis for evaluating Tikhonov-regularized marching computations. With standard marching procedures, relatively high values of the regularization parameter r are found to be necessary; the resulting loss of resolution leads to erroneous solutions. When slowly divergent schemes are used, much lower values of r are possible, leading to reasonably accurate reconstruction of thermal histories at the active surface.