Efficient Solution of Nonlinear Ordinary Differential Equations Expressed in S-system Canonical Form

Abstract

Ordinary differential equations can be recast into a nonlinear canonical form called an S-system. Evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions, recognized as among the most general, are special cases of S-systems. Identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations. By capitalizing on the regular structure of S-systems, efficient formulas for a variable-order, variable-step Taylor-series method are developed. The minimum improvement in efficiency over conventional Runge–Kutta methods is shown to be more than 20 to 60 percent for equations expressed in S-system form. Tests with implemented methods demonstrate that 10- to 100-fold reductions in time are actually realized for solution of S-systems. To examine the potential benefit of recasting ordinary differential equations and then solving with ESSYNS (Evaluation and Simulation of Synergistic Systems), the Taylor-series method for S-systems, a standard set of nonstiff to moderately stiff benchmark problems is recast into S-system form. Each problem is solved in original form with conventional Runge–Kutta, Adams, and Gear methods and in S-system form with ESSYNS. Solving in S-system form with ESSYNS typically requires fewer function evaluations and is often faster overall than solving in original form with conventional methods. At stringent error tolerances, ESSYNS can be 10 to 20 times faster. ESSYNS also is shown to have the advantages of predictable error control, high accuracy based in part on insensitivity to roundoff error, and robustness including the ability to effectively treat discontinuities in derivatives.