Overview¶

NodePy (Numerical ODEs in Python) is a Python package for designing, analyzing, and testing numerical methods for initial value ODEs. Its development was motivated by my own research in time integration methods for PDEs. I found that I was frequently repeating tasks that could be automated and integrated. Initially I developed a collection of MATLAB scripts, but this became unwieldy due to the large number of files that were necessary and the more limited capability for code reuse.

NodePy represents an object-oriented approach, in which the basic object is a numerical ODE solver. The idea is to design a laboratory for such methods in the same sense that MATLAB is a laboratory for matrices. Some distinctive design goals are:

• Plug-and-play: any method can be applied to any problem using the same syntax. Also, properties of different kinds of methods are available through the same syntax. This makes it easy to compare different methods.
• Abstract representations: Generally, the most abstract (hence powerful) representaton of an object is used whenever possible. Thus, order conditions are generated using products on rooted trees (or other recursions) rather than being hard-coded.
• Floating-point representation: Most quantities, such as coefficients of integration methods, are represented with double-precision numbers. This means also that, for the most part, properties of methods that are in truth exact (discretely-valued) are determined by numerical calculations, using appropriate tolerances. Thus the “order” of a method is determined by checking whether the order conditions are satisfied to within a small value (near machine-epsilon). Of course, for some purposes an exact representation of rational coefficients and exact computation of method properties would be preferable. However, since such an approach would be less generally applicable, this is planned as future work.

In general, user-friendliness of the interface and readability of the code are prioritized over performance.

NodePy includes capabilities for applying the methods to solve systems of ODEs. This is mainly intended for testing and comparison; for realistic problems of interest in most fields, time-stepping in Python will be too slow. One way around this is to wrap Fortran or C functions representing the right-hand-side of the ODE, and we are looking into this.