A numerical tour of wave propagation

Discretization of NPML

Note that all sub-equations can be formulated in the following form:

$$\frac{\partial f}{\partial t}+d f=\gamma.$$ (49)

The analytic solution of this equation is

$$f=-\frac{1}{d}e^{-d t}+\frac{1}{d}\gamma$$ (50)

In discrete form,

$$\begin{split}f(k\Delta t)=-\frac{1}{d}e^{-dk\Delta t}+\frac{1... ...rac{1}{d}e^{-d t}e^{-dk\Delta t}+\frac{1}{d}\gamma. \end{split}$$ (51)

Thus,

$$f((k+1)\Delta t)=e^{-d\Delta t}f(k\Delta t)+\frac{1}{d}(1-e^{-d\Delta t})\gamma$$ (52)

For $\frac{\partial \Omega_{xx}}{\partial t}+d(x)\Omega_{xx}=d(x)\frac{\partial\tau_{xx}}{\partial x}$ , $\gamma=d(x)\frac{\partial\tau_{xx}}{\partial x}$ , the update rule becomes

$$\Omega_{xx}^{k+1}=e^{-d(x)\Delta t}\Omega_{xx}^{k}+(1-e^{-d(x)\De... ...rtial x}=b_x\Omega_{xx}^{k}+(1-b_x)\frac{\partial\tau_{xx}^{k+1/2}}{\partial x}$$ (53)

where $b_x=e^{-d(x)\Delta t}$ and $b_z=e^{-d(z)\Delta t}$ . $\Omega_{xx}$ , $\Omega_{xz}$ , $\Omega_{zx}$ , $\Omega_{zz}$ , $\Psi_{xx}$ , $\Psi_{xz}$ , $\Psi_{zx}$ and $\Psi_{zz}$ can be obtained in the same way:

$$\left\{ \begin{split}\Omega_{xx}^{k+1}=b_x\Omega_{xx}^{k}+(1-b_... ...b_z)\frac{\partial v_z^{k+1/2}}{\partial z}\\ \end{split} \right.$$ (54)

As can be seen from Eq. (27), we only need to subtract the reflection part $\Omega$ and $\Psi$ after global updating (Eq. (9)). We summarize this precedure as follows:

Step 1: Perform the computation of Eq. (9) in whole area;

Step 2: In PML zone, subtract decaying parts according to Eq. (27).

A numerical tour of wave propagation