Automated PDE Solving in Julia with MethodOfLines.jl

If you want to simulate something, sooner or later you’re going to come across partial differential equations. But solving PDEs is hard, right? It doesn’t have to be! In this talk we'll cut to the chase: how do I copy paste the textbook description of my PDE into Julia symbolic syntax and get a solution? MethodOfLines.jl is the answer, and in this talk we'll show you how to do it!

MethodOfLines.jl is a system for the automated discretization of symbolically defined partial differential equations (PDEs), by the method of lines. By recognizing different linear and nonlinear terms in the specified system, we build a performant semidiscretization by symbolically applying effective finite difference schemes, which we then used to generate optimized Julia code. Consequently, one can solve the system with an appropriate ordinary differential equation (ODE) solver.

In this 30 minute talk, the audience will learn how to use MethodOfLines.jl to discretize and solve an example PDE that represents a physical simulation which arises in research, gaining the knowledge and skills to apply these tools to their own problems. They will also learn about some of the internals of MethodOfLines.jl. This will arm them with the knowledge required to implement improved finite difference schemes, benefiting their own research and others in the community. Finally, we will outline the proposed direction of development for the package moving forwards.