Interactive solution of partial differential equations by the Method-of-lines

Abstract

A convenient, generally applicable technique for programming Method-of-lines solutions of linear or nonlinear partial differential equations employs a new implementation of W. Schiesser's general-purpose differentiation operators. A novel vector compiler reads a vector equation or differential equation and produces efficient code for n corresponding scalar equations without causing any runtime loop overhead. Simulations can combine partial and ordinary differential equations. Programs compile and run immediately on a mouse click to permit truly interactive modeling and simulation. Originally designed for experiments with Monte Carlo simulation, neural networks, and fuzzy logic, the new runtime compiler easily generates many different Method-of-lines algorithms for partial-differential equation systems. As a simple example, we exhibit the complete solution of a heat-conduction problem with one space dimension.