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In order to make a solve a conservation law, need to construct instances of the following types:
gather them in a ConservedSystem and Simulator object:
system = ConservedSystem(equation, reconstruction, numerical_flux, timestepper)
simulator = Simulator(system, grid, t0)
and solve in place using simulate! or simulate_and_aggregate!:
simulate!(simulator, T, max_dt, callbacks)
Example
For example, we can solve Burgers' equation with Neumann boundary conditions and Riemann initial condition on a uniform grid with \(N=20\) cells spanning \((x_L, x_R) = (0,1)\). The simplest way to do this is is with no reconstruction, a forward Euler time-stepper, and for example the Lax-Friedrichs numerical flux.
eq = BurgersEQ()
bc = NeumannBC()
N = 20
x_L, x_R = 0.0, 1.0
x = cell_centers(N, x_L, x_R)
u0(x) = x .< 0.5 ? 1.0 : 0.0
U0 = u0.(x)
grid = UniformGrid1D(N, bc, U0, (x_L, x_R))
F = LaxFriedrichsFlux()
reconstruction = NoReconstruction()
timestepper = ForwardEuler(grid)
we gather these into a System object, and solve using a Simulator object.
system = ConservedSystem(eq, reconstruction, F, timestepper)
simulator = Simulator(system, grid, 0.)
U, t = simulate_and_aggregate!(simulator, 1., 0.1)
Next, we can animate the solution
u(x, t) = u0(x-0.5t)
Viz.animate_solution(U', u, grid, t, 2)
Extending the functionallity
The package homemade_conslaws is designed to be easily extensible, making heavy use of multiple dispatch. This is done by implementing the abstract types in the table above.