DiffEqOperators.jl

DiffEqOperators.jl is a package for finite difference discretization of partial differential equations. It serves two purposes:

1. Building fast lazy operators for high order non-uniform finite differences.
2. Automated finite difference discretization of symbolically-defined PDEs.

Note: (2) is still a work in progress!

For the operators, both centered and upwind operators are provided, for domains of any dimension, arbitrarily spaced grids, and for any order of accuracy. The cases of 1, 2, and 3 dimensions with an evenly spaced grid are optimized with a convolution routine from NNlib.jl. Care is taken to give efficiency by avoiding unnecessary allocations, using purpose-built stencil compilers, allowing GPUs and parallelism, etc. Any operator can be concretized as an Array, a BandedMatrix or a sparse matrix.

Documentation

For information on using the package, see the stable documentation. Use the in-development documentation for the version of the documentation which contains the unreleased features.

Example 1: Automated Finite Difference Solution to the Heat Equation

```julia using OrdinaryDiffEq, ModelingToolkit, DiffEqOperators, DomainSets

Parameters, variables, and derivatives

@parameters t x @variables u(..) Dt = Differential(t) Dxx = Differential(x)^2

1D PDE and boundary conditions

eq = Dt(u(t,x)) ~ Dxx(u(t,x)) bcs = [u(0,x) ~ cos(x), u(t,0) ~ exp(-t), u(t,Float64(pi)) ~ -exp(-t)]

Space and time domains

domains = [t ∈ Interval(0.0,1.0), x ∈ Interval(0.0,Float64(pi))]

PDE system

@named pdesys = PDESystem(eq,bcs,domains,[t,x],[u(t,x)])

Method of lines discretization

dx = 0.1 order = 2 discretization = MOLFiniteDifference([x=>dx],t;centered_order=order)

Convert the PDE problem into an ODE problem

prob = discretize(pdesys,discretization)

Solve ODE problem

sol = solve(prob,Tsit5(),saveat=0.1) ```

Example 2: Finite Difference Operator Solution for the Heat Equation

```julia using DiffEqOperators, OrdinaryDiffEq

# Heat Equation

This example demonstrates how to combine OrdinaryDiffEq with DiffEqOperators to solve a time-dependent PDE.

We consider the heat equation on the unit interval, with Dirichlet boundary conditions:

∂ₜu = Δu

u(x=0,t) = a

u(x=1,t) = b

u(x, t=0) = u₀(x)

For a = b = 0 and u₀(x) = sin(2πx) a solution is given by:

u_analytic(x, t) = sin(2πx) * exp(-t(2π)^2)

nknots = 100 h = 1.0/(nknots+1) knots = range(h, step=h, length=nknots) ordderiv = 2 ordapprox = 2

const Δ = CenteredDifference(ordderiv, ordapprox, h, nknots) const bc = Dirichlet0BC(Float64)

t0 = 0.0 t1 = 0.03 u0 = u_analytic.(knots, t0)

step(u,p,t) = Δbcu prob = ODEProblem(step, u0, (t0, t1)) alg = KenCarp4() sol = solve(prob, alg) ```