Timefractional Diffusion

This repository provides a MFEM based implementation to simulate the time-fractional heat equation $$((\partial_t)^\alpha - \nabla \cdot D \nabla) u = 0$$ with homogenous Dirichlet BC. We use MFEM for the spatial discretization and to compute the mass and stiffness matrix. For the time integration, we employ the modified Crank-Nicholson algorithm introduce by Khristenko and Wohlmuth (2021) (Lemma 5.6). The algorithm is enclosed in a simple for loop.

Dependencies

The project has a few dependencies:

• MFEM: finite element software needed for spatial discretization. Available via their GitHub
• CNPY: The project requires the adaptive Antoulas-Anderson (AAA) algorithm which is available for C++ and we had to fall back to Python. The results are saved in the .npy format and later read with CNPY in the C++ code.
• Python: We use a conda environment to execute the python scripts. The environment can be found in py_scripts/conda_environment.yml and be installed with conda env create -f py_scripts/conda_environment.yml
• We make use of a library called mittag_leffler available via GitHub that should be copied into the project directory, i.e. timefractional-diffusion/mittag_leffler. We had to remove some . from the imports in the library to get the scripts in py_scripts running.
• glVis for visualization of u(t,x) available via GitHub.

Setting up the conda environment

conda env create -f py_scripts/conda_environment.yml
conda list | grep fractional
conda activate fractional

Running the python scripts

There are two python scripts aaa_weights.py and visualize_ML.py that both should be executed while being in the project directory. The former computes coefficients via the AAA algorithm, the latter visualizes the analytic 1D solution. For more infos, consult the scripts or their help messages.

python py_scripts/aaa_weights.py --help
python py_scripts/aaa_weights.py --alpha 0.5 --visualize
python py_scripts/visualize_ML.py --help
python py_scripts/visualize_ML.py --alpha 0.5

Simulating the Timefractional-Heat equation

mkdir build && cd build
cmake .. && make -j <num_processors>
ctest  # run the few available tests to avoid bad surprises
./tf-diffusion --help # see available parameters
./tf-diffusion -d 1 -r 7 -mi 500 # run 1D simulation 

Visualization

Start glvis in server mode to listen before running ./tf-diffusion.

./glvis/build/glvis

Over all workflow

python py_scripts/aaa_weights.py --alpha 0.5 # compute expansion coefficients
./glvis/build/glvis # start glvis in different terminal
cd build
./tf-diffusion -d 1 -r 7 -mi 500 # solve PDE numerically with coefficients
cd ..
python py_scripts/visualize_ML.py --alpha 0.5 # compare glVis with analytic sol.