Fast simulation of shock evolution in multirelaxing fluids using intrinsic coordinates

Discretization of thin shocks requires high computational cost for conventional time-domain algorithms that solve the augmented Burgers equation describing nonlinear propagation in a relaxing fluid. The Burgers equation can be expressed using intrinsic coordinates [Hammerton and Crighton, J. Fluid Mech. 252, 585–599 (1993)], in which waveforms remain single valued beyond the point at which waveform steepening renders them multivalued in physical coordinates. This coordinate transformation underlies a simulation approach in which shocks are inserted into the multivalued pressure waveform using the equal area rule from weak shock theory, yielding a single-valued waveform in physical coordinates without requiring fine discretization of the shock. A two-stage approach is developed by first solving the Burgers equation in intrinsic coordinates out to distances where shocks are no longer expected to be thin. Beyond that distance, the solution can be propagated in physical coordinates using a time-domain algorithm without restrictively small discretization. The two-stage approach significantly reduces computational cost compared to time-domain numerical solutions using exclusively physical coordinates and maintains high accuracy. This advantage is demonstrated with simulations of N waves propagating in air and shocks with exponential tails in seawater, each involving two relaxation mechanisms [Willis et al., JASA 157, 3824–3835 (2025)].