CPML

Convolutional Perfectly Matched Layer (CPML) is an absorbing boundary condition used in time-domain simulations of wave propagation, most notably in computational electromagnetics. It serves as an improvement over the original perfectly matched layer by providing superior absorption of outgoing waves across a broad range of frequencies and incidence angles, including grazing incidence. CPML achieves this by implementing a stretched-coordinate transformation in time domain, realized through convolution rather than purely frequency-domain operations.

In practical terms, CPML replaces simple damping layers with an anisotropic, lossy boundary region. The method introduces auxiliary variables that represent the temporal convolution of field components with decaying kernels. These auxiliary equations are updated during each time step, allowing the boundary to absorb energy effectively without requiring explicit time-domain convolutions. The CPML parameters — such as the conductivity grading, layer thickness, and polynomial order — are selected to minimize reflections, often with conductivities increasing gradually toward the exterior.

CPML is widely used with finite-difference time-domain (FDTD) and other time-domain methods and is applicable to isotropic and anisotropic materials. It has been extended beyond electromagnetics to other wave systems, including acoustics and elastodynamics, where similar absorbing boundary concepts apply. Advantages over traditional PML include improved absorption performance for a wide range of frequencies and angles, and robust stability when the CPML parameters are chosen suitably. Limitations involve added implementation complexity and memory overhead due to the auxiliary variables, as well as potential parameter sensitivity in highly dispersive or intricate geometries.

Overall, CPML is a standard technique for simulating open or unbounded domains in wave physics, helping to minimize artificial reflections in finite computational domains.