Taming Explosive Computational Instability: Compensated Explicit Time-Marching Schemes in Multidimensional, Nonlinear, Well-Posed or Ill-Posed Initial Value Problems for Partial Differential Equations.

One hundred years ago, Richardson introduced the second order accurate explicit `leapfrog' centered time difference scheme, and used it in his failed attempt at numerical weather prediction. This attractive computational scheme is unconditionally unstable in well-posed parabolic problems. Likewise, the attractive first order accurate pure explicit scheme requires a stringent Courant stability restriction on the time step, and is seldom used. Instead, stable implicit schemes such as Crank-Nicolson, requiring solution of large scale algebraic problems at each time step, are widely used.

For ill-posed initial value problems, all stepwise time-marching difference schemes, whether explicit or implicit are unconditionally unstable.

This talk discusses methods for stabilizing the leapfrog and pure explicit schemes, by applying compensating smoothing operators at each time step to quench the instability. In spatial domains for which the eigenvalues and eigenfunctions of the Laplacian operator are known, such smoothing operators are readily synthesized. In particular, highly efficient FFT algorithms can be used in rectangular regions. Such smoothing induces a distortion away from the true solution. This is the "stabilization penalty". However, that error is small enough in many problems of interest to allow for useful results. The stabilized leapfrog and pure explicit schemes can be applied in many well-posed multidimensional problems on fine meshes, without the need to solve algebraic systems at each step, a very significant advantage.

The above stabilized schemes can also be run backward in time, and can solve previously intractable multidimensional nonlinear ill-posed time-reversed parabolic equations, as well as other time-reversed problems such as coupled sound and heat flow, and wave propagation in viscous fluids. Unexpectedly, the first order accurate pure explicit scheme is found to be better behaved in time-reversed problems, than is the second order accurate leapfrog scheme. Instructive illustrative examples will be given.

Speaker Bio: Alfred S. Carasso received the Ph.D degree in mathematics at the University of Wisconsin in 1968. He was a professor of mathematics at the University of New Mexico, and a visiting staff member at the Los Alamos National Laboratory, prior to joining the National Institute of Standards and Technology in 1982. His major research interests lie in the theoretical and computational analysis of ill-posed continuation problems in partial differential equations, together with their application in inverse heat transfer, system identification, non-destructive evaluation,and image reconstruction. He is the author of original theoretical papers, is a patentee in the field of image processing, and is an active speaker at national and international conferences in applied mathematics.

Presentation Slides: PDF

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