DiffEqOperators.jl is a package for finite difference discretization of partial differential equations. It serves two purposes:
- Building fast lazy operators for high order non-uniform finite differences.
- 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
BandedMatrix or a sparse matrix.
Example 1: Automated Finite Difference Solution to the Heat Equation
using OrdinaryDiffEq, ModelingToolkit, DiffEqOperators # Parameters, variables, and derivatives @parameters t x @variables u(..) @derivatives Dt'~t @derivatives Dxx''~x # 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 ∈ IntervalDomain(0.0,1.0), x ∈ IntervalDomain(0.0,Float64(pi))] # PDE system pdesys = PDESystem(eq,bcs,domains,[t,x],[u]) # Method of lines discretization dx = 0.1 order = 2 discretization = MOLFiniteDifference(dx,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
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(2*π*x) * exp(-t*(2*π)^2) nknots = 100 h = 1.0/(nknots+1) knots = range(h, step=h, length=nknots) ord_deriv = 2 ord_approx = 2 const Δ = CenteredDifference(ord_deriv, ord_approx, h, nknots) const bc = Dirichlet0BC(Float64) t0 = 0.0 t1 = 0.03 u0 = u_analytic.(knots, t0) step(u,p,t) = Δ*bc*u prob = ODEProblem(step, u0, (t0, t1)) alg = KenCarp4() sol = solve(prob, alg)