Parker-Sochacki method

In mathematics, the Parker-Sochacki method is an algorithm for solving systems of ordinary differential equations (ODEs), which has been developed by G. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces Maclaurin series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form.

Summary

The Parker-Sochacki method rests on two simple observations:

• If a set of ODEs has a particular form, then the Picard method can be used to find their solution in the form of a power series.
• If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original ODEs.

Several coefficients of the power series are calculated in turn, a time step is chosen, the series is evaluated at that time, and the process repeats.

The end result is a high order piecewise solution to the original ODE problem. The order of the solution desired is an adjustable variable in the program that can change between steps. The order of the solution is only limited by the floating point representation on the machine running the program.

Advantages

The method requires only addition, subtraction, and multiplication, making it very convenient for high-speed computation. (The only divisions are inverses of small integers, which can be precomputed.) Use of a high order--calculating many coefficients of the power series--is convenient. (Typically a higher order permits a longer time step without loss of accuracy, which improves efficiency.) The order and step size can be easily changed from one step to the next. It is possible to calculate a guaranteed error bound on the solution. Arbitrary precision floating point libraries allow this method to compute arbitrarily accurate solutions.

Disadvantages

Most methods for numerically solving ODEs require only the evaluation of derivatives for chosen values of the variables, so systems like MATLAB include implementations of several methods all sharing the same calling sequence. Users can try different methods by simply changing the name of the function called. The Parker-Sochacki method requires more work to put the equations into the proper form, and cannot use the same calling sequence.

External links

• Polynomial ODEs - Examples, Solutions, Properties, retrieved 2010 . A thorough explanation of the paradigm and application of the Parker-Sochacki method
• Joseph W. Rudmin (1998), Application of the Parker-Sochacki Method to Celestial Mechanics, retrieved November 11, 2013 . A demonstration of the theory and usage of the Parker-Sochacki method, including a solution for the classical Newtonian N-body problem with mutual gravitational attraction.
• The Modified Picard Method., retrieved November 11, 2013 . A collection of papers and some Matlab code.