Walk-on-Spheres PDE Solver

This repo implements a minimal walk-on-spheres-based PDE solver for various elliptic PDEs on arbitrary input 2D or 3D meshes.

Dependencies

• C++17 compiler
• CMake >= 3.20
• MPI (e.g. Open MPI, MPICH, Cray MPI)
• HDF5 with parallel support
• Python >= 3.10 with numpy, h5py, matplotlib (visualisation only)

MacOS

brew install cmake open-mpi hdf5-mpi

HPC

Load the appropriate modules, for example:

module load cmake cray-hdf5-parallel

Submit using a slurm script.

Build

cmake -B build
cmake --build build -j

Run

mpirun -np <ranks> ./build/wos [equation] [Nx Ny] [Nz] [mesh.obj]

All arguments are positional and optional. The dimension of the mesh is auto-detected from the file. Pass an unknown equation name to get the list of registered equations.

Examples:

# 2D screened Poisson on the annulus, 64x64 grid, 8 ranks
mpirun -np 8 ./build/wos screened_poisson 64 64 meshes/annulus.obj

# 3D screened Poisson on the sphere, 32x32x32 grid
mpirun -np 8 ./build/wos screened_poisson 32 32 32 meshes/sphere.obj

Output is written to wos.h5 in the current directory.

Visualise

pip install -r requirements.txt
python plot.py --mesh meshes/annulus.obj wos.h5

The --mesh argument must match the mesh used to produce the .h5 file. The output file is u.png by default.

Adding a new equation

Each equation lives in its own src/equations/<name>.cpp and is registered in src/equations/equations.hpp. When creating a new equation, the relevant files need to be added to the CMakeLists.txt.