A Brief Overview of Solvers and Solver Options

Note: This document assumes that the reader is somewhat familiar with numerical methods used to solve ODE's. For more information, consult a numerical analysis or scientific computing textbook.

Runge-Kutta-Fehlberg 4th/5th

Runge-Kutta_fehlberg 4th/5th is an adaptive stepsize algorithm. It adjusts the stepsize in such a way as to keep the absolute error in a step less than Absolute Tol. and the fractional error less than Relative Tol. See Numerical Recipies in C: The Art of Scientific Computing, by Press, Teutolsky, Vetterling, and Flannery, for more information on the algorithm.

Absolute Tol.: RKF45 adjusts the stepsize in such a way as to keep the absolute error in a step less than Absolute Tol.
Relative Tol.: RKF45 adjusts the stepsize in such a way as to keep the fractional error in a step less than Relative Tol.
Min Step Size : The minimum stepsize that RKF45 should attempt to use. If the requested error tolerances cannot be achieved even when using the minumum stepsize, the solver will report an error.
Max Step Size : The maximum stepsize that RKF45 should attempt to use.
Points/Time: The number of points that should be graphed per unit time.

Rosenbrock

The Rosenbrock method is an adaptive, semi-implicit, multi-stage method . The Rosenbrock method can be thought of as a generalization of the Runge-Kutta methods. This is done by adding more parameters into the equations for this method. As a result of this generalization, the Rosenbrock Solver can solve stiff differential equations. This is something that the Euler Method and the Runge-Kutta methods cannot do well. Like the RKF45 method described above, it adjusts the stepsize in such a way as to keep the absolute error in a step less than Absolute Tol. and the fractional error less than Relative Tol.

Rosenbrock Options: See the description of the RKF45 options above.

Euler

The Euler algorithm is a very basic first-order, fixed step size ODE solver. It is included in ODEToolkit merely for demonstration purposes as it is inaccurate.

Euler Options:

Points/Time: Specified the number of output points per unit time that the solver should compute. The stepsize is computed by dividing the solve span by Points/Time.