Using the same ideas as for the parabolic equation, hyperbolic implements the numerical solution of
for x in Ω, with the initial conditions
for all x in Ω, and usual boundary conditions. In particular, solutions of the equation utt - cΔu = 0 are waves moving with speed .
Using a given triangulation of Ω, the method of lines yields the second order ODE system
with the initial conditions
after we eliminate the unknowns fixed by Dirichlet boundary conditions. As before, the stiffness matrix K and the mass matrix M are assembled with the aid of the function assempde from the problems
–∇ · (c∇u) + au = f and –∇ · (0∇u) + du = 0.