FaADE.jl
A Summation by Parts code for solving the field aligned Anisotropic Diffusion Equation.
Features
FaADE.jl is a code for solving the field aligned anisotropic diffusion equation
\[\frac{\partial u}{\partial t} = \nabla\cdot(\mathbf{K} \nabla ) u\]
where
\[\mathbf{K} = k_\perp\mathbf{I} + (k_\parallel - k_\perp)\frac{\mathbf{B}\mathbf{B}^T}{\|\mathbf{B}\|^2}.\]
- Uses the Summation by Parts formulation with Simultaneous Approximation Terms (SBP-SAT) for boundary conditions [1].
- Parallel penalty operator used to apply diffusion along vector field lines.
- Currently arbitrary parallel mappings can be provided in Cartesian geometry or an ODE for mapping grid points along field lines.
- Provides solutions in 1D for:
- diffusion problems
- with a parallel map in the second dimension
- and solutions in 2D for:
- diffusion problems
- with a parallel map in the third dimension
Examples
The best place to start is in the Examples section in the navigation bar to the left.
Modules
- Simultaneous Approximation Terms
- Boundary operators
- Helpers
- Derivative operators
- First derivative operators
- Second Derivative operators
- Internals
- Time Integration
Similar software
- SummationByParts.jl: A Julia implementation of a wide range of SBP operators
- pyoculus: A magnetic field diagnostic package in python based on an earlier FORTRAN implementation oculus
References
The mathematical background for this package can be found in:
- D. Muir, K. Duru, M. Hole, and S. Hudson, “An efficient method for the anisotropic diffusion equation in magnetic fields,” Feb. 08, 2023, arXiv. doi: 10.48550/arXiv.2303.15447.
- D. Muir, K. Duru, M. Hole, and S. Hudson, “A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields,” Apr. 09, 2024, arXiv. doi: 10.48550/arXiv.2306.00423.