SplitFXM 0.2

1D Finite-Difference/Volume Split Newton Solver

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 two and for ease of communication, let's refer to first set of variables as "outer" and the second as "inner".

• Holding the outer variables fixed, Newton iteration is performed till convergence using the sub-Jacobian

• One Newton step is performed for the outer variables with inner held fixed (using its sub-Jacobian)

• This process is repeated till convergence criterion is met for the full system (same as in Newton)

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.schemes import default_scheme
from splitfxm.visualize import draw

# Define the problem
method = 'FDM'
m = AdvectionDiffusion(c=0.2, nu=0.001, method=method)
d = Domain.from_size(20, 1, 1, ["u", "v", "w"]) # nx, nb_left, nb_right, variables
ics = {"u": "gaussian", "v": "rarefaction"}
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(dt=0.1)
bounds = [[-1., -2., 0.], [5., 4., 3.]]
iter = s.steady_state(split=True, split_loc=1, bounds=bounds)

# Visualize
draw(d, "label")

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	20 seconds
Sparse	~0.6 seconds

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

python -m pytest -s benchmark

Whom to contact?

Please direct your queries to gpavanb1 for any questions.

Acknowledgements

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