Numerical solution of nonlinear differential equations with algebraic contraints II: practical implications

In this paper we investigate the behavior of numerical ODE methods for the solution of systems of differential equations coupled with algebraic constraints. Systems of this form arise frequently in the modelling of problems from physics and engineering; we study some particular examples from fluid dynamics and constrained mechanical systems. We investigate some of the practical difficulties of implementing variable-stepsize backward differentiation formulas for the solution of these equations, showing how to overcome problems of matrix ill-conditioning, and giving convergence tests and error tests which are supported by theory.

[1]  Edwin Hewitt,et al.  Foundations of Modern Analysis (vol. X of Pure and Applied Mathematics). , 1961 .

[2]  A. Sluis.  Condition, equilibration and pivoting in linear algebraic systems , 1970 .

[3]  C. W. Gear,et al.  Simultaneous Numerical Solution of Differential-Algebraic Equations , 1971 .

[4]  J. Baumgarte.  Stabilization of constraints and integrals of motion in dynamical systems , 1972 .

[5]  Documentation for DFASUB: a program for the solution of simultaneous implicit differential and nonlinear equations , 1973 .

[6]  J. Wittenburg,et al.  Dynamics of systems of rigid bodies , 1977 .

[7]  N. Orlandea,et al.  A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 2 , 1977 .

[8]  R. Sani,et al.  On the time-dependent solution of the incompressible Navier-Stokes equations in two and three dimensions , 1980 .

[9]  R. Sani,et al.  Solution of the time-dependent incompressible Navier-Stokes equations via a penalty Galerkin finite element method , 1981 .

[10]  Solving the Navier-Stokes equations with LSODI and the method of lines , 1981 .

[11]  L. Petzold.  A description of dassl: a differential/algebraic system solver , 1982 .

[12]  L. Petzold.  Differential/Algebraic Equations are not ODE's , 1982 .

[13]  K. Brenan,et al.  Difference approximations for higher index differential-algebraic systems with applications in trajectory control , 1984, The 23rd IEEE Conference on Decision and Control.

[14]  C. W. Gear,et al.  ODE METHODS FOR THE SOLUTION OF DIFFERENTIAL/ALGEBRAIC SYSTEMS , 1984 .

[15]  Per Lötstedt,et al.  Numerical solution of nonlinear differential equations with algebraic constraints I: Convergence results for backward differentiation formulas , 1986 .