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A numerical solver for conservation laws based on central schemes

Project description

centpy

Central schemes for conservation laws in Python.

The schemes are translated into Python from CentPack written by Jorge Balbás and Eitan Tadmor.

Usage

Centpy provides to the user three main classes for parameters, equations, and solvers. Examples of instances for parameters and equations are in tests/example_parameters.py and tests/example_equations.py.

The numerical solution of a one-dimensional Burgers equation is discussed below.

Parameters

The parameter classes are simple data classes without methods: Pars1d and Pars1d defined in parameters.py. Each attribute has a default variable, but it is recommended that all attributes are set explicitly. The attributes are:

Attribute Description
x_init left grid point
x_final right grid point
t_final evolution time
dt_out time step of storage
J number of interior grid points
cfl CFL number
scheme solver scheme (fd2, sd2, or sd3)

An instance of the parameter class can be created as follows.

pars_burgers1d = centpy.Pars1d(
    x_init=0.0,
    x_final=2.0 * np.pi,
    t_final=10,
    dt_out=0.05,
    J=400,
    cfl=0.75,
    scheme="sd3")

Note that the parameter data class does not have a member for the time step dt, because it is calculated dynamically during the solution of the equation based on the CFL number and maximum spectral radius.

Equations

The equations are abstract base classes which require methods for setting initial data, boundary conditions, fluxes, and spectral radius. Additional helper methods and parameters can be added depending on the problem. The equations class inherits all attributes of the parameters class. The space-time grid is constructed in this step based on the parameters. The Burgers equation class is defined below.

class Burgers1d(centpy.Equation1d):
    def initial_data(self):
        return np.sin(self.x) + 0.5 * np.sin(0.5 * self.x)

    def boundary_conditions(self, u):
        u[0] = u[-4]
        u[1] = u[-3]
        u[-2] = u[2]
        u[-1] = u[3]

    def flux_x(self, u):
        return 0.5 * u * u

    def spectral_radius_x(self, u):
        return np.abs(u)

The boundary conditions are periodic, so the data on the ghost points are copied from the interior points on the opposite end.

Solution

There are two solver classes: Solver1d and Solver2d defined in solver1d.py and solver2d.py respectively. To construct the solution, we create an instance of the Burgers1d class with the parameters, and give the equation instance as input to the solver class.

eqn_burgers1d = Burgers1d(pars_burgers1d)
soln_burgers = centpy.Solver1d(eqn_burgers1d)
soln_burgers.solve()

After the solver step, the instance soln_burgers includes the solution array u_n. Depending on the shape of the array, plots and animations can be easily constructed. Examples are given in the animations notebook tests/animations.ipynb.

The options for the central solver are fd2 for second order fully-discrete method, sd2 for second order semi-discrete method, and sd3 for third order semi-discrete method. Information about these solvers is given at the appendix of the CentPack User Guide.

LaTeX formulas and animations for the examples are given in the Jupyter notebook tests/animations.ipynb.

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