PML

Solving wave equation in time/frequency domain with PML requires decent modifications on equation

$\frac{\partial w}{\partial t}=Hw$

with initial Cauchy conditions and $H$ is anti-Hermitian. Where $w$ is phase space coordinate. Anti-Hermitian operator brings oscillations and numerically makes the problem not easy to be accurate.

The idea of PML is to transform:

$\frac{\partial }{\partial x}\to \frac{1}{1+i\frac{\sigma (x)}{\omega }}\frac{\partial }{\partial x}$

within PML region, using positive $\sigma$ to damp the energy. In other regions, $\sigma =0$. However, a dilemma is:

• We would like to make PML region as small as possible to reduce overall cost.

• We would like $\sigma$ as large as possible but continuous, then it would absorb as much as possible.

However, in order to preserve the thinness of our layer, then $\sigma$ is chosen as a rapid increasing function from domain of interest into PML, numerically it will require finer mesh to reduce reflection error from discretization, that is not a cheap deal. A slow increasing function may not need such a fine mesh due to Adiabatic, but it may not be able to absorb the desired level of energy.

As we quoted from Steve Johnson

With PML, however, the constant factor is very good to start with, so experience shows that a simple quadratic or cubic turn-on of the PML absorption usually produces negligible reflections for a PML layer of only half a wavelength or thinner.

It has been seen that the layer need not be thick, half a wavelength is enough, however, that also requires resolution. Usually within a wavelength, there should be 10~20 pixels or nodes, which means the resolution $h\sim 0.1\lambda$.

In frequency domain, solving Helmholtz equation

$\Delta u+k^{2}u=0$

is hard for large wave number $k$ because of its oscillatory operator kernel. PML is a method as described above. Using limit absorption principle is another way. Instead of solving original equation, adding absorption term with remove the non-uniqueness of the equation.

$\Delta u+k^{2}u+ik\sigma u=0$

with conductivity $\sigma >0$. And let $\sigma \to 0^{+}$ will force the solution approximate the true solution.

Consider a combination of the two. If our median has conductivity to make wave dissipative. And we also want no-reflection effect to ruin the result, we may use PML to solve the equation. Boundary condition can be vanishing, since positive $\sigma$ will act as a damping factor on the solution.