DJ4Earth / Burgers.jl

Sensitivities of the Burgers equation

DiffDistPDE

DiffDistPDE is an implementation of a differentiable distributed PDE solver in Julia. It is developed using the latest language features and packages of Julia, which required ample workarounds and is meant as a demonstration of future capabilities for developing differentiable numerical simulation software. It uses

• MPI.jl for distributed computing,
• Enzyme.jl for differentiation,
• and Checkpointing.jl for reducing the memory footprint of the differentiated code.

Quickstart

Example usage of the solver is given in the examples folder. The following example solves the 2D Burgers equation with Dirichlet boundary conditions

$$u(t, x, −L) = u(t, x, L) = u(t, −L, y) = u(t, L, y) = 0$$

and initial condition

$$u(0, x, y) = exp(−x^2 − y^2), v(0, x, y) = exp(−x^2 − y^2)$$

include("examples/burgers.jl")

MPI.Init()
scaling = 10

Nx = 100*scaling
Ny = 100*scaling
tsteps = 1000*scaling

μ = 0.01 # # U * L / Re,   nu

dx = 3e-2
dy = 3e-2
dt = 1e-3 # dt < 0.5 * dx^2

snaps = 1000
rank = MPI.Comm_rank(MPI.COMM_WORLD)
if rank == 0
    println(
        "Running Burgers with Nx = $Nx, Ny = $Ny, tsteps = $tsteps,
        μ = $μ, dx = $dx, dy = $dy, dt = $dt, snaps = $snaps"
    )
end
burgers(Nx, Ny, tsteps, μ, dx, dy, dt, snaps; writedata = true)

The following files will be written

• IC.jld - Initial condition
• final.jld - Final state
• adjoints.fld - Adjoint state of the final energy with respect to the initial state

The plots can be generated using plotdata/plot_teaser.jl.

Please note that the solver is not optimized for performance and is not meant for production use. It is meant as a demonstration of Julia for developing differentiable numerical simulation software.