Engquist–Majda absorbing boundary condition

Theory

A simple form of Engquist–Majda absorbing boundary condition can be formulated through one-dimensional scalar wave equation:

${\displaystyle \left(L_x^2-v^{-2}{L}_t^2\right)U(x,t)=GU(x,t)=0}$

where ${\displaystyle L_{\{x,t\}}={\mathrm{\partial }}_{\{x,t\}}}$ are the partial differential operators and ${\displaystyle v}$ is the wave velocity; ${\displaystyle G}$ is the resultant wave function operator. This operator can be split into two one-way wave operators, resulting in advection equations:

${\displaystyle GU(x,t)=G^{+}{G}^{-}U(x,t)=0}$

${\displaystyle G^{+}=L_x+v^{-1}{L}_t}$

${\displaystyle G^{-}=L_x-v^{-1}{L}_t}$

Applying the following boundary conditions at a simulation region from ${\displaystyle x=0}$ to ${\displaystyle x=L}$ results in the cancellation of outgoing waves:

${\displaystyle G^{+}U(x,t){|}_{x=0}=\left(L_x+v^{-1}{L}_t\right)U(x,t){|}_{x=0}=0}$

${\displaystyle G^{-}U(x,t){|}_{x=L}=\left(L_x-v^{-1}{L}_t\right)U(x,t){|}_{x=L}=0}$

In the case of higher dimensions, i.e. the two-dimensional scalar wave equation, the operator form can be written as:

${\displaystyle \left(L_x^2+L_y^2-v^{-2}{L}_t^2\right)U(x,y,t)=G^{+}{G}^{-}U(x,y,t)=0}$

For waves traveling along the ${\displaystyle x}$-axis, the operator splitting results in:

${\displaystyle G^{+}U(x,y,t){|}_{x=0}=\left(L_x+v^{-1}{L}_t\sqrt{1-v^2\frac{L_y^2}{L_t^2}}\right)U(x,t){|}_{x=0}=0}$

${\displaystyle G^{-}U(x,y,t){|}_{x=L}=\left(L_x-v^{-1}{L}_t\sqrt{1-v^2\frac{L_y^2}{L_t^2}}\right)U(x,t){|}_{x=L}=0}$

The terms in square brackets constitute pseudo-differential operators and their theoretical exact forms are nonlocal. In practice, they can be locally approximated with their Taylor series representations. First-order approximation ${\displaystyle (\sqrt{1-s^2}\approx 1)}$ yields:

${\displaystyle G^{+}\approx L_x+v^{-1}{L}_t}$

${\displaystyle G^{-}\approx L_x-v^{-1}{L}_t}$

which is equivalent to the Engquist–Majda condition for one-dimensional wave equation: in the theoretical limit, it perfectly absorbs the normally incident waves while causing reflections for waves impinging on other angles. A second-order approximation ${\displaystyle (\sqrt{1-s^2}\approx 1-s^2/2)}$ yields a better performance:

${\displaystyle G^{+}\approx L_x+v^{-1}{L}_t\left[1-v^2\frac{L_y^2}{2L_t^2}\right]=L_x+v^{-1}{L}_t-\frac{v^2{L}_y^2}{2L_t^2}}$

${\displaystyle G^{-}\approx L_x-v^{-1}{L}_t\left[1-v^2\frac{L_y^2}{2L_t^2}\right]=L_x-v^{-1}{L}_t+\frac{v^2{L}_y^2}{2L_t^2}}$

Higher-order boundary conditions can also be obtained through Padé or Chebyshev approximation of the pseudo-differential operators; these are known as generalized Trefethen–Halpern absorbing boundary conditions.