Solution of Second Order Differential Equations
Using the Godunov Integration Method
Abstract
This MS Thesis proposes a method for the direct numerical solution of systems
of second order differential equations. Many continuous systems can naturally
be expressed in terms of second order differential equations. Problems in
mechanics, hydrodynamics, and electrical circuits fall in this category. Most
numerical integration techniques currently in use rely on separating each 2nd
order equation into two 1st order equations. This thesis investigates an
alternative integration technique proposed by Godunov which solves second order
differential equations directly. It is hoped that the Godunov method may
provide an advantage over 1st order system solvers in the areas of numerical
stability domains, truncation errors and the number of floating point operations
required for a solution. A numerical stability domain will be developed for
the Godunov method both analytically and numerically, and expressions will be
developed for the truncation error which results from using the Godunov method
to solve the wave equation, a mechanical model of the human body and a 7th order
passive electrical circuit. Analytic results will then be compared against
numerical solutions of these three systems. It will be shown that the Godunov
method compares favorably against the Adams-Bashforth 3rd order method when used
to solve both the mechanical system and the hyperbolic partial differential
equation, but that there are potential problems when this method is used to
simulate electrical circuits which result in integro-differential equations.
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