SplitFXM

1D Finite-Difference or Finite-Volume using asymmetric stencils with adaptive mesh refinement and steady-state solver using Newton and Split-Newton approach

What does 'split' mean?

The system is divided into multiple segments, and for ease of communication, let’s refer to the first segment of variables as "outer" and the remaining as "inner".

• Holding the outer variables fixed, Newton iteration is performed recursively for the inner variables, using the sub-Jacobian associated with them, until convergence is reached.

• One Newton step is then performed for the outer variables, while the inner variables are kept fixed, using the sub-Jacobian for the outer subsystem.

• This process is repeated, alternating between solving the inner and outer subsystems, until the convergence criterion for the entire system (similar to standard Newton) is met.

Example:

Consider a system of 5 variables, with the split locations at indices [1, 4]. This results in the following segments:

• a1 (variables from 0 to 1)
• a2 a3 a4 (variables from 1 to 4)
• a5 (variable at index 4)

1. First, the innermost segment a5 is solved recursively using Newton's method while holding the variables a1 and a2 a3 a4) fixed. This step is repeated until the convergence criterion for a5 is met.

2. Next, one Newton step is taken for the segment a2 a3 a4, with a5 held fixed. This step is followed by solving a5 again till convergence.

3. This alternating process repeats: solving for a5 until convergence, then one step for a2 a3 a4, and so on, until all subsystems converge.

Finally, one Newton step is performed for a1, with the other segments fixed. This completes one cycle of the split Newton process.

How to install and execute?

Just run

pip install splitfxm

There is an examples folder that contains a test model - Advection-Diffusion

You can define your own equations by simply creating a derived class from Model and adding to the _equations using existing or custom equations!

A basic driver program is as follows

from splitfxm.domain import Domain
from splitfxm.simulation import Simulation
from splitfxm.models.advection_diffusion import AdvectionDiffusion
from splitfxm.schemes import default_scheme
from splitfxm.visualize import draw
import matplotlib.pyplot as plt

# Define the problem
method = 'FVM'
m = AdvectionDiffusion(c=0.2, nu=0.001, method=method)
# nx, nb_left, nb_right, variables
d = Domain.from_size(20, 1, 1, ["u", "v", "w"])
ics = {"u": "gaussian", "v": "rarefaction", "w": "tophat"}
bcs = {
    "u": {
        "left": "periodic",
        "right": "periodic"
    },
    "v": {
        "left": {"dirichlet": 3},
        "right": {"dirichlet": 4}
    },
    "w": {
        "left": {"dirichlet": 2},
        "right": "periodic"
    }
}
s = Simulation(d, m, ics, bcs, default_scheme(method))


# Advance in time or to steady state
s.evolve(t_diff=0.1)
bounds = [[-1., -2., -2.], [5., 4., 3.]]
iter = s.steady_state(split=True, split_locs=[1, 2], bounds=bounds)

# Visualize
draw(d, "label")
plt.show()

How to build from source?

Since v0.4.0, SplitFXM utilizes Cython for accelerated computation. To build from source, you will need to install Cython and run the following command:

python setup.py build_ext --inplace

Run benchmark

There is a benchmark that is included, which compares the time it takes to generate both a sparse and dense Jacobian. The results are as follows:

For N=250,

Method	Time
Dense	45 seconds
Sparse	~0.6 seconds

The benchmark can be executed from the parent folder using the command

python -m pytest -s benchmark

How to run tests?

To run the tests, execute the following command from the parent folder:

python -m pytest tests

You can use the -s flag to show print outputs of the tests

How to get coverage?

To get coverage, execute the following command from the parent folder:

python -m pytest --cov=splitfxm --cov-report <option> tests

The option can be related to showing covered/missed lines or specifying the output format of the report. For example, to get a line-by-line report, use the following command:

python -m pytest --cov=splitfxm --cov-report term-missing tests

Whom to contact?

Please direct your queries to gpavanb1 for any questions.

You can also reach out to us on the SplitFXM Google Group

Acknowledgements

Special thanks to Cantera and WENO-Scalar for serving as an inspiration for code architecture.

Citing

If you are using SplitFXM in any scientific work, please make sure to cite as follows

@software{pavan_b_govindaraju_2025_14827049,
  author       = {Pavan B Govindaraju},
  title        = {gpavanb1/SplitFXM: v0.5.0},
  month        = feb,
  year         = 2025,
  publisher    = {Zenodo},
  version      = {v0.5.0},
  doi          = {10.5281/zenodo.14827049},
  url          = {https://doi.org/10.5281/zenodo.14827049},
}