Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

This book is about solving differential equations numerically using integration techniques that preserve as many of the geometric and physical properties of the underlying systems as possible. (The word “geometric” in the title is intended to include all kinds of structural and physical properties, but there is nothing inherently geometrical in the integration methods.) The structure-preserving algorithms that the authors study were first considered in the 1980’s. Their primary goal is to identify long-term behavior of the solutions and to identify - when possible - conservation laws or other qualitative features of the dynamics. A related goal is to handle highly oscillatory problems, where the systems are periodic or quasi-periodic and require solutions over time intervals that include a very large number of periods. In either case one desires numerically accurate values of the solutions at times that may be far in the future.

 

Good general-purpose tools for the numerical solution of many differential equations have been developed over the last several years and are widely available. They provide a collection of reliable, easy-to-use software for both initial and boundary value problems. The current book goes beyond them to consider problems where such software is no longer useful. 

 

This is an advanced and specialized text. A prototype for the kind of problems it considers is in celestial mechanics where solutions are required over very large time intervals. (For example, when one attempts to answer questions about the long-term stability of the solar system.) In this case, the differential equations define a symplectic system in which physical properties of energy and momentum must be conserved at each stage of a numerical solution algorithm. In other applications, conservation of volume or energy may be important.

 

Systems with highly oscillatory behavior arise in current treatments of classical and quantum mechanics, astrophysics, and molecular dynamics. Conventional methods often don’t work well in these circumstances, so the usual software packages are inadequate. The main difference between the approach taken by classical software packages and the newer methods described in this book is that the new techniques are more closely tailored to the problem at hand so as to capture as much as possible of its unique structure.

 

The book has two parts. The first considers solutions of highly oscillatory ordinary differential equations; the second treats oscillatory problems for partial differential equations. With ordinary differential equations, many variations of methods based on Runge-Kutta or Runge-Kutta-Nyström techniques are employed. The partial differential equations of primary interest are nonlinear wave equations including variations of the Klein-Gordon equation.

 

 Each part has discussions of numerical experiments and examples. The numerical examples show the preliminaries for computations, describe the method, and then present numerical results. Specific algorithms and code are not included, but many references are provided.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.