Example of 1D Wave Equation with Absorbing Boundary Condition

wave.jl

# solve 1D wave equation with ABC
using PyPlot
using Plots
using LinearAlgebra


N = 100
x = LinRange(-1.,1.,N+1)
Δx = x[2]-x[1]
NT = 200
Δt = 2.0/NT

U = zeros(N+1, NT+2)

U[:,1] = exp.(-10x.^2)
U[:,2] = exp.(-10x.^2)

cfun = x->1. 
c = cfun.(x)


for i = 1:NT
    U[2:end-1,i+2] = 2*U[2:end-1, i+1] - U[2:end-1, i] + (Δt/Δx)^2 * c[2:end-1] .* (
        U[3:end, i+1] + U[1:end-2,i+1] - 2U[2:end-1,i+1]
    )
    r = cfun(x[end]-Δx/2)*Δt/Δx
    U[end, i+2] = U[end-1, i+1] + (r-1)/(r+1)*(U[end-1,i+2]-U[end,i+1])

    r = cfun(x[1]+Δx/2)*Δt/Δx
    U[1, i+2] = U[2, i+1] - (1-r)/(1+r)*(U[2,i+2] - U[1, i+1])
end

@gif for i = 2:2:NT+2
    Plots.plot(x, U[:,i],ylims=(0.0,0.8), label="wave at t=$(round((i-2)*Δt, digits=2))")
end

This example shows a simple finite difference scheme for 1D wave equation with obsorbing boundary condition. In a future gist, we will try to solve the inverse problem: given u(x,t=0) and u(x,t=1), we want to recover c(x).

Reference: 1D FDTD