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Alfred S. Carasso studies an inverse problem for nonlinear advection-diffusion equations: finding initial conditions that will evolve toward specified target data. An explicit stabilized leapfrog scheme is run backward in time to construct candidate initial values, which are then evolved forward to test the result.
The numerical experiments use sharply varying grayscale image data to expose both successful and unsuccessful cases. Equations, error tables and visual comparisons show how the result depends on the time interval and on coefficients in the evolution equation. The complete report emphasizes the feasibility limits of this ill-posed problem as well as the practical value of a fast computational method.
Alfred S. Carasso (2022). Data Assimilation in 2D Nonlinear Advection Diffusion Equations, Using an Explicit O(Δt)² Stabilized Leapfrog Scheme Run Backward in Time. National Institute of Standards and Technology. https://doi.org/10.6028/NIST.TN.2227. Republished courtesy of the National Institute of Standards and Technology.