PyClaw tutorial: Solve the acoustics equations

PyClaw is designed to solve general systems of hyperbolic PDEs of the form

\[\begin{equation} \kappa(x) q_t + A(q,x) q_x = 0. \end{equation}\]

As an example, in this tutorial we’ll set up a simulation that solves the acoustics equations in one dimension:

\[\begin{split}\begin{eqnarray} &p_t + K u_x = 0\\ &u_t + \frac{1}{\rho} p_x = 0 \end{eqnarray}\end{split}\]

The key to solving a particular system of equations with PyClaw or other similar codes is a Riemann solver. Riemann solvers for many systems are available as part of the clawpack/riemann package.

We’ll assume that you’ve already followed the Installing PyClaw instructions.

The following instructions show how to set up a problem step-by-step in an interactive shell. See acoustics_1d for the full source on which this is based.

The commands below should be typed at the Python prompt; we recommend using IPython.

>>> from clawpack import pyclaw
>>> from clawpack import riemann

The Solver

PyClaw includes various algorithms for solving hyperbolic PDEs; each is implemented in a Solver object. So the first step is to create a solver

>>> solver = pyclaw.ClawSolver1D(riemann.acoustics_1D)

Next we set the boundary conditions. We’ll use a wall (wall) condition at the left boundary and a non-wall (zero-order extrapolation) condition at the right boundary

>>> solver.bc_lower[0] = pyclaw.BC.wall
>>> solver.bc_upper[0] = pyclaw.BC.extrap

The domain

Next we need to set up the grid. We do so by defining the physical domain and the computational resolution. We’ll use the interval \((-1,1)\) and 200 grid cells:

>>> domain = pyclaw.Domain([-1.0], [1.0], [200])

Notice that the calling sequence is similar to numpy’s linspace command.

Finally, we set up a Solution object, which will hold the solution values:

>>> solution = pyclaw.Solution(solver.num_eqn, domain)

Initial condition

Now we will set the initial value of the solution

>>> state = solution.state
>>> xc = state.grid.p_centers[0]      # Array containing the cell center coordinates
>>> from numpy import exp
>>> state.q[0,:] = exp(-100 * (xc-0.75)**2) # Pressure: Gaussian centered at x=0.75.
>>> state.q[1,:] = 0.                       # Velocity: zero.

Problem-specific parameters

The acoustics equations above have some coefficients – namely, the bulk modulus \(K\) and density \(\rho\) – that must be defined. Furthermore, checking the code for the Riemann solver we’ve chosen reveals that it expects us to provide values for the impedance \(Z\) and sound speed \(c\). These values are stored in a Python dictionary called problem_data that is a member of the State

>>> from math import sqrt
>>> rho = 1.0
>>> bulk = 1.0
>>> state.problem_data['rho'] = rho
>>> state.problem_data['bulk'] = bulk
>>> state.problem_data['zz'] = sqrt(rho*bulk)
>>> state.problem_data['cc'] = sqrt(bulk/rho)

The controller

The most convenient way to run a PyClaw simulation is by using a Controller object. The controller directs the solver in advancing the solution and handles output.

>>> controller = pyclaw.Controller()
>>> controller.solution = solution
>>> controller.solver = solver
>>> controller.tfinal = 1.0

At last everything is set up! Now run the simulation

>>> status =

This should print out a few lines indicating the output times. It also prints the minimum and maximum tipe-step used, the number of steps required for the computation and the maximum CFL number. The simplest way to plot the solution is

>>> from clawpack.pyclaw import plot
>>> plot.interactive_plot() 

That’s it! Your first PyClaw simulation. Of course, we’ve only scratched the surface of what PyClaw can do, and there are many important options that haven’t been discussed here. To get an idea, take a look through the pyclaw/examples directory and try running some other examples. It’s also a good idea to get more deeply acquainted with the main Understanding Pyclaw Classes.