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Finite volume discretizations for Python

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Creating finite volume equation systems with ease.

pyfvm provides everything that is needed for setting up finite volume equation systems. The user needs to specify the finite volume formulation in a configuration file, and pyfvm will create the matrix/right-hand side or the nonlinear system for it. This package is for everyone who wants to quickly construct FVM systems.


Linear equation systems

pyfvm works by specifying the residuals, so for solving Poisson's equation with Dirichlet boundary conditions, simply do

import meshplex
import meshzoo
import numpy as np
from scipy.sparse import linalg

import pyfvm
from pyfvm.form_language import Boundary, dS, dV, integrate, n_dot_grad

class Poisson:
    def apply(self, u):
        return integrate(lambda x: -n_dot_grad(u(x)), dS) - integrate(lambda x: 1.0, dV)

    def dirichlet(self, u):
        return [(lambda x: u(x) - 0.0, Boundary())]

# Create mesh using meshzoo
vertices, cells = meshzoo.rectangle_tri(
    np.linspace(0.0, 2.0, 401), np.linspace(0.0, 1.0, 201)
mesh = meshplex.Mesh(vertices, cells)

matrix, rhs = pyfvm.discretize_linear(Poisson(), mesh)

u = linalg.spsolve(matrix, rhs)

mesh.write("out.vtk", point_data={"u": u})

This example uses meshzoo for creating a simple mesh, but anything else that provides vertices and cells works as well. For example, reading from a wide variety of mesh files is supported (via meshio):

mesh ="pacman.e")

Likewise, PyAMG is a much faster solver for this problem

import pyamg

ml = pyamg.smoothed_aggregation_solver(matrix)
u = ml.solve(rhs, tol=1e-10)

More examples are contained in the examples directory.

Nonlinear equation systems

Nonlinear systems are treated almost equally; only the discretization and obviously the solver call is different. For Bratu's problem:

import pyfvm
from pyfvm.form_language import *
import meshzoo
import numpy as np
from sympy import exp
import meshplex

class Bratu:
    def apply(self, u):
        return integrate(lambda x: -n_dot_grad(u(x)), dS) - integrate(
            lambda x: 2.0 * exp(u(x)), dV

    def dirichlet(self, u):
        return [(u, Boundary())]

vertices, cells = meshzoo.rectangle_tri(
    np.linspace(0.0, 2.0, 101), np.linspace(0.0, 1.0, 51)
mesh = meshplex.Mesh(vertices, cells)

f, jacobian = pyfvm.discretize(Bratu(), mesh)

def jacobian_solver(u0, rhs):
    from scipy.sparse import linalg

    jac = jacobian.get_linear_operator(u0)
    return linalg.spsolve(jac, rhs)

u0 = np.zeros(len(vertices))
u = pyfvm.newton(f.eval, jacobian_solver, u0)

mesh.write("out.vtk", point_data={"u": u})

Note that the Jacobian is computed symbolically from the Bratu class.

Instead of pyfvm.newton, you can use any solver that accepts the residual computation f.eval, e.g.,

import scipy.optimize

u = scipy.optimize.newton_krylov(f.eval, u0)

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