Space marching difference schemes in the nonlinear inverse heat conduction problem

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Abstract
For ill-posed initial value problems, step by step marching computations are unconditionally unstable, and necessarily blow-up numerically as the mesh is refined. However, for the 1D nonlinear inverse heat conduction problem, the author shows how to construct consistent marching schemes that blow-up much more slowly than the counterpart analytical problem. Several new space marching finite difference schemes are formulated and compared with existing schemes relative to their error amplification properties. Using the Lax-Richtmyer theory, the author evaluates the L2 norms of the linearized discrete solution operators mapping the sensor data into the desired temperature and gradient histories at the inaccessible active surface. Various combinations of space and time differencing are examined, leading to 18 different algorithms.

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