# DiffEqOperators.jl

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 `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

```
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)
```