B.Sc. Thesis

The Predicted Sequential Regularization Method for Differential-Algebraic Equations

Keywords: sequential regularization methods, predicted sequential regularization methods, Runge-Kutta methods, differential-algebraic equations, method of lines.


Differential-algebraic equations (DAEs) can be used to describe the evolution of many interesting and important systems in a variety of disciplines. Common examples include multibody systems as an aspect of mechanical engineering, biomechanics, or robotics; Navier-Stokes equations for incompressible fluid dynamics; and Kirchoff's laws from electrical engineering applications. Most of these problems present particular analytical and numerical difficulties due to the algebraic constraints that distinguish DAEs from ordinary differential equations.

This thesis involves the implementation of two recent numerical methods for solving DAEs: the Sequential Regularization Method (SRM) and the related Predicted Sequential Regularization Method (PSRM). The resulting implementation breaks new ground by the addition of automatic step-size control code to the SRM. This is an important step in the evolution of the SRM/PSRM from experimental code to useful software.

It is shown that although all DAEs can be converted into ODEs, it is usually not desirable to do so because the numerical stability of the system is undermined in the process. The techniques of regularization and stabilization are discussed as they play an important role in the SRM/PSRM. The derivation of the SRM/PSRM and the associated algorithms are covered in some detail. The implementation is discussed including a section on how to use the software to solve additional problems.

The theoretical orders of the SRM and PSRM are tested on several problems with known solutions. Finally, comparisons are made between existing SRM codes and the new implementation as well as experimental comparisons between the SRM and PSRM.