First-Order Perfectly Matched Layer

In physics, the one-dimensional wave equation describes wave propagation along a single spatial direction. When mathematically modeling this phenomenon, partial differential equations are typically employed to capture the wave dynamics. To enhance the solution accuracy, a Perfectly Matched Layer (PML) can be incorporated into the computational domain. The PML serves as an artificial absorbing layer that effectively reverses wave propagation direction, preventing unwanted reflections at domain boundaries and eliminating interference patterns. From an implementation perspective, the PML can be programmed by introducing complex coordinate stretching factors to the wave equation, which creates exponential decay within the boundary layer without generating numerical reflections. Key implementation aspects include properly tuning the PML attenuation profile and ensuring smooth transition between the main computational domain and the PML region through gradient-based absorption coefficients.