Solution of Stiff Systems Described by Ordinary Differential Equations by Means of Regression Backward Difference Formulae (RBDF)

Abstract

In the past research has been done to solve stiff systems described by ordinary differential equations (ODEs). An important result are the famous Backward Difference Formulae (BDF). These methods are capable of solving stiff ODE-systems up to accuracy order six - accuracy order six means that the error made at each integration step is roughly proportional to the seventh power of the step-size. So far, no BDF algorithms of seventh order and higher, have been found that are stable.

This thesis proposes the Regression Backward Difference Formulae (RBDF) as new numerical solution methods for stiff systems described by first order ordinary differential equations. The RBDF algorithms derived in this thesis, by means of a new regression technique, will be of sixth and seventh order, and it will be shown that some of the sixth order RBDF algorithms compare favorably against the sixth order BDF.

The results for the new seventh order RBDF algorithms are shown, but not compared to BDF since no stable seventh order BDF technique exists.

It can be expected that RBDF methods of order higher than seven may be found by using the proposed regression approach.

In particular, celestial analysis demands highly accurate calculation and integration. Therefore, this might be one area where even higher order RBDF techniques than seventh order RBDF could be applied.

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Last modified: February 6, 2006 -- © François Cellier