Conservative Eulerian numerical methods for geometrical shock dynamics (GSD) and detonation shock dynamics (DSD)

G. B. Whitham's geometrical shock dynamics (GSD) is a useful reduced-order model of shock front propagation in an inert medium. GSD derives an "area-Mach number" state equation which has been shown to be equivalent to an intrinsic relation between the shock front normal velocity, normal acceleration, and curvature. This type of relationship is also observed in advanced detonation shock dynamics (DSD) modeling of high explosives. Existing numerical methods for solving the differential system of GSD with arbitrary area-Mach number relations include Lagrangian wavefront tracking (ray tracing), paraxial approximations (space marching), and non-conservative fast marching methods (FMM). We present here an iterative, conservative Eulerian method for viscosity-limiting solutions of GSD and advanced DSD that overcomes difficulties faced by previous methods, such as merging shock fronts, multiple source/inlet directions, and correct Rankine-Hugoniot jump conditions. Several tests are detailed that demonstrate the accuracy and efficiency of this new method.