Numerical questions in ODE boundary value problems

Michael Robert Osborne


Classical considerations of stability in ODE initial and boundary problems are mirrored by corresponding properties (stiff stability, di-stability) in problem discretizations. However, computational categories are not precise, and qualitative descriptors such as `of moderate size' cannot be avoided where size varies with the sensitivity of the Newton iteration in nonlinear problems, for example. Sensitive Newton iterations require close enough initial estimates. The main tool for providing this in boundary value problems is continuation with respect to a parameter. If stable discretizations are not available, then adaptive meshing is needed to follow rapidly changing solutions. Use of such tools can be necessary in stable nonlinear situations not covered by classical considerations. Implications for the estimation problem are sketched. It is shown how to choose appropriate boundary conditions for the embedding method. The simultaneous method is formulated as a constrained optimization problem. It avoids explicit ODE characterization and appears distinctly promising. However, its properties are not yet completely understood.

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