PhysicalFFT.jl

FFT PDE solvers in Julia language.

WARNING: This package is under development!!!

]add PhysicalFDM

or

using Pkg; Pkg.add("PhysicalFDM")

or

using Pkg; Pkg.add("https://github.com/JuliaAstroSim/PhysicalFDM.jl")

To test the Package:

]test PhysicalFDM

This package is extracted from AstroNbodySim.jl. You may find more advanced examples there.

You can also reference the usage of finite differencing solver PhysicalFDM.jl.

The main disadvantage of PhysicalFDM.jl is that the computational complexity (the matrix size) scales with $M^{d^d}$, where $M$ is the mesh size in each direction and d is the dimension of the problem. Consequently, for meshes $\ge 16^3$, the memory usage ($\ge$ 64GB) and computation time ($\ge$ 10 hrs) are not affordable.

PhysicalFFT.jl supports resolution of $512^3$ with minimum effort: 10 sec on both GPU (shared GPU memory is used) and CPU. For meshes smaller than $256^3$, the computation time on GPU is 1~4 orders of magnitude shorter than on CPU. However, the vacuum boundary conditions, which are necessary for isolated gravitational systems, are not yet supported in PhysicalFFT.jl. Nevertheless, the errors from periodic boundary conditions are tolerable if the simulation box is sufficiently large compared to the system's length scale.

$$\Delta u = f$$

using PhysicalFFT
using PhysicalFFT.PhysicalMeshes
using PhysicalMeshes.PhysicalParticles

sol(p::PVector) =  sin(2*pi*p.x) * sin(2*pi*p.y) * sin(2*pi*p.z) + sin(32*pi*p.x) * sin(32*pi*p.y) * sin(2*pi*p.z) / 256
init_rho(p::PVector) = -12 * pi * pi * sin(2*pi*p.x) * sin(2*pi*p.y) * sin(2*pi*p.z) - 12 * pi * pi * sin(32*pi*p.x) * sin(32*pi*p.y) * sin(32*pi*p.z)

function test_fft3D(Nx, boundary=Periodic())
    m = MeshCartesianStatic(;
        xMin = 0.0,
        yMin = 0.0,
        zMin = 0.0,
        xMax = 1.0,
        yMax = 1.0,
        zMax = 1.0,
        Nx = Nx - 1,
        Ny = Nx - 1,
        Nz = Nx - 1,
        NG = 0,
        dim = 3,
        boundary,
    )
    m.rho .= init_rho.(m.pos)
    fft_poisson!(m, m.rho, m.config.boundary)
    s = sol.(m.pos)
    r = m.phi .- s

    return L2norm(r)
end

test_fft3D(8, Periodic())

• Vacuum boundary conditions for isolated system
• Test GPU
• FFT spectral solver for Schrödinger-Poisson equation (SPE)

• Basic data structure: PhysicalParticles.jl
• File I/O: AstroIO.jl
• Initial Condition: AstroIC.jl
• Parallelism: ParallelOperations.jl
• Trees: PhysicalTrees.jl
• Meshes: PhysicalMeshes.jl
• Finite differencing solver PhysicalFDM.jl
• FFT solver PhysicalFFT.jl
• Plotting: AstroPlot.jl
• Simulation: AstroNbodySim.jl