PyModPDE: A python software for modified equation analysis

Abstract

The modified equation is a useful tool in the analysis of numerical methods for partial differential equations (PDEs). It gives insight into the stability, diffusion, and dispersion properties of a given numerical scheme. Its derivation, however, is rather tedious and error-prone due to the enormous amount of algebra involved. PyModPDE is a python software that uses a novel approach to generate the modified equation. It takes a discrete PDE as its input and outputs the modified equation in LaTeX format. We discuss the novel approach on which PyModPDE is based and then validate the software using one and two-dimensional PDEs. PyModPDE serves as an essential tool for computational scientists and engineers for both educational and research purposes.

Keywords

• Modified equation analysis
• Finite difference schemes
• Stability analysis
• Numerical methods
• Error analysis