Adaptive Partitioning Techniques for Ordinary Differential Equations

Many stiff systems of ordinary differential equations (ODEs) modeling practical problems can be partitioned into loosely coupled subsystems. In this paper the objective of the partitioning is to permit the numerical integration of one time step to be performed as the solution of a sequence of small subproblems. This reduces the computational complexity compared to solving one large system and permits efficient parallel execution under appropriate conditions. The subsystems are integrated using methods based on low order backward differentiation formulas. This paper presents an adaptive partitioning algorithm based on a classical graph algorithm and techniques for the efficient evaluation of the error introduced by the partitioning. The power of the adaptive partitioning algorithm is demonstrated by a real world example, a variable step-size integration algorithm which solves a system of ODEs originating from chemical reaction kinetics. The computational savings are substantial. AMS subject classification (2000): 65L06, 65Y05.