Wolfram Language

Partial Differential Equations

Solve a Wave Equation with Absorbing Boundary Conditions

Solve a 1D wave equation with absorbing boundary conditions.

Specify a wave equation with absorbing boundary conditions. Note that the Neumann value is for the first time derivative of .

In[1]:=
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eqn = D[u[t, x], {t, 2}] == D[u[t, x], {x, 2}] + NeumannValue[-Derivative[1, 0][u][t, x], x == 0 || x == 1];

Specify initial conditions for the wave equation.

In[2]:=
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u0[x_] := Evaluate[D[0.125 Erf[(x - 0.5)/0.125], x]]; ic = {u[0, x] == u0[x], Derivative[1, 0][u][0, x] == 0};

Solve the equation using the finite element method.

In[3]:=
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ufun = NDSolveValue[{eqn, ic}, u, {t, 0, 1}, {x, 0, 1}, Method -> {"MethodOfLines", "SpatialDiscretization" -> {"FiniteElement"}}];

Visualize the wave equation with absorbing boundary conditions.

In[4]:=
Click for copyable input
list = Table[ Plot[ufun[t, x], {x, 0, 1}, PlotRange -> {-0.1, 1.3}], {t, 0, 1, 0.1}]; ListAnimate[list]
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