A High-Order Time and Space Formulation of the Unsplit Perfectly Matched Layer for the Seismic Wave Equation Using Auxiliary Differential Equations (ADE-PML)

Unsplit convolutional perfectly matched layers (CPML) for the velocity and stress formulation of the seismic wave equation are classically computed based on a second-order finite-difference time scheme. However it is often of interest to increase the order of the time-stepping scheme in order to increase the accuracy of the algorithm. This is important for instance in the case of very long simulations. We study how to define and implement a new unsplit non-convolutional PML called the Auxiliary Differential Equation PML (ADE-PML), based on a high-order Runge-Kutta time-stepping scheme and optimized at grazing incidence. We demonstrate that when a second-order time-stepping scheme is used the convolutional PML can be derived from that more general non-convolutional ADE-PML formulation, but that this new approach can be generalized to high-order schemes in time, which implies that it can be made more accurate. We also show that the ADE-PML formulation is numerically stable up to 100,000 time steps.