2.2.7 Klein-Gordon equation

This example solves the Klein-Gordon equation in one dimension, showing the effect of the mass term on wave propagation. With mass=0 the equation reduces to the standard wave equation; increasing the mass introduces dispersion.

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from pde import KleinGordonPDE, ScalarField, UnitGrid

grid = UnitGrid([128])  # generate grid
u = ScalarField.from_expression(grid, "exp(-((x - 32) / 5) ** 2)")

eq = KleinGordonPDE(speed=1, mass=0.5)  # define the pde
state = eq.get_initial_condition(u)
result = eq.solve(state, t_range=50, dt=0.01)
result[0].plot()