Hybrid Asynchronous Perfectly Matched Layer for seismic wave propagation in unbounded domains

Reflection error analysis for wave propagation problems solved by a heterogeneous asynchronous time integrator

“…The reader may refer to Appendix C for the proof of (32). Using the continuity of the Lagrange multiplier at times T k+1 and T k−1 , respectively, one may notice that the Lagrange multipliers on the right-hand side of Equation (32) are linked by the following identity:…”

“…Combining (32), (34), and (35), we obtain the second relation between the amplitudes A h v , R h v , and T h v as follows:…”

“…Proceeding by analogy with the explicit/implicit case, the relations between the wave amplitudes and the Lagrange multipliers, deduced straightforwardly from Equation (14) and the proof of the first relation in (32), are given by…”

“…Despite this limitation, the GC method permits the use of much higher time‐step ratios (from 100 to 1000) when applied to structural dynamic problems . The efficiency of the method has also been tested in previous works for wave propagation problems involving multi–time‐step absorbing layers . Further enhancements with regard to the numerical dissipation have been carried out by Prakash and Hjelmstad to build an energy‐conserving hybrid multi–time‐step method (PH method).…”

“…[27][28][29] The efficiency of the method has also been tested in previous works for wave propagation problems involving multi-time-step absorbing layers. [30][31][32] Further enhancements with regard to the numerical dissipation have been carried out by Prakash and Hjelmstad 33 to build an energy-conserving hybrid multi-time-step method (PH method). Recently, a new second-order accuracy method (also energy conserving) has been developed by Mahjoubi et al, 34 labeled the MGC method (for Mahjoubi, Gravouil, and Combescure) and able to couple Newmark, HHT-, 35 Simo, 36 Krenk,37 and Verlet 38 time integrators.…”