The Discretized Linear Wave

The linear wave equation in one dimension has the following Hamiltonian (see e.g. [2]):

\[ \mathcal{H}_\mathrm{cont}(q, p; \mu) := \frac{1}{2}\int_\Omega \mu^2(\partial_\xi q(t, \xi; \mu))^2 + p(t, \xi; \mu)^2 d\xi,\]

where the domain is $\Omega = (-1/2, 1/2)$. We then divide the domain into $\tilde{N}$ equidistantly spaces points^[1] $\xi_i = i\Delta_\xi - 1/2$ for $i = 1, \ldots, \tilde{N}$ and $\Delta_xi := 1/(\tilde{N} + 1)$.

\[ \mathcal{H}_h(z) = \sum_{i = 1}^{\tilde{N}}\frac{\Delta{}x}{2}\left[ p_i^2 + \mu^2 \frac{(q_i - q_{i - 1})^2 + (q_{i+1} - q_i)^2}{2\Delta{}x} \right].\]

The discretized linear wave equation example of an completely-integrable system, i.e. a Hamiltonian system evolving in $\mathbb{R}^{2n}$ that has $n$ Poisson-commuting invariants of motion (see [3]).

For evaluating the system we specify the following initial^[2] and boundary conditions:

\[\begin{aligned} q_0(\omega;\mu) := & q(0, \omega; \mu) \\ p(0, \omega; \mu) = \partial_tq(0,\xi;\mu) = & -\mu\partial_\omega{}q_0(\xi;\mu) \\ q(t,\omega;\mu) = & 0, \text{ for } \omega\in\partial\Omega. \end{aligned}\]

By default GeometricProblems uses the following parameters:

import GeometricProblems.LinearWave as lw

lw.default_parameters

(μ = 0.6, N = 256)

And if we integrate we get:

problem = lw.hodeproblem()
sol = integrate(problem, ImplicitMidpoint())

# plot 6 time steps
time_steps = 0 : (length(sol.t) - 1)  ÷ 5 : (length(sol.t) - 1)
p = plot()
for time_step in time_steps
    plot!(p, lw.compute_domain(lw.Ñ + 2), sol.q[time_step, :], label = "t = "*string(round(sol.t[time_step]; digits = 2)))
end

p

As we can see the thin pulse travels in one direction.

Library functions