UC Irvine and MIT
Partial differential equations (PDEs) are used throughout scientific disciplines, modeling diverse phenomena such as the spread of chemical concentrations across biological organisms to global temperature flows. However, solving PDEs efficiently is not easy: it requires a vertical toolkit with many interconnected pieces. In this workshop we will introduce the participants to some basic PDEs, where they come from, and how to tie together the various tools across the Julia package ecosystem to solve them efficiently.
We will start by focusing on elliptic problems and show how these decompose into solving sparse linear systems. To solve the resulting linear systems, the participants will be introduced three methods: Julia’s special matrix types for efficient dense solutions, BandedMatrices.jl for more generic banded operators, and IterativeSolvers.jl for Krylov methods. The differences between the methodologies and the current status of distributed and GPU compatibility will be discussed. From there, the extension to nonlinear elliptic problems will be shown as effectively solving nonlinear systems with sparse Jacobians, and demonstrations/comparisons of Roots.jl, NLsolve.jl, and Sundials.jl’s KINSOL for solving the resulting systems will be addressed. Lastly, the concept of pseudospectral discretizations will be introduced using the library ApproxFun.jl, it will be shown how this methodology simply leads to different linear/nonlinear systems.
After understanding the tooling for elliptic PDEs, “time-dependent” PDEs such as parabolic and hyperbolic PDEs will be introduced. It will be shown how similar discretizations as done in the elliptic portion lead to systems of coupled ordinary differential equations (ODEs). It will be demonstrated how to efficiently solve the resulting ODE systems via DifferentialEquations.jl using methods such as via banded Jacobian Rosenbrock integrators, Newton-Krylov BDF, Implicit-Explicit (IMEX) Runge-Kutta, and exponential integrators like ETDRK4. Participants will be shown how to integrate pseudospectral operators from ApproxFun.jl and linear solvers from IterativeSolvers.jl to customize the integrators to their problem. Additionally, special time-stepping issues for hyperbolic PDEs and the strong-stability preserving (SSP) integrators will be introduced.
Together, the workshop participants should be able to leave with a good understanding of how to tie together the Julia scientific packages to efficiently solve a large class of partial differential equations.
I am a mathematician and theoretical biologist at the University of California, Irvine. My programming language of choice is Julia and I am the lead developer of the JuliaDiffEq organization dedicated to solving differential equations (and includes the package DifferentialEquations.jl). My research is in time stepping methods for solving stochastic differential equations (SDEs) and applications to stochastic partial differential equations (SPDEs) which model biological development.