A parallel space-time $p$-adaptive discontinuous Galerkin method for nonlinear acoustics

In this paper, we introduce and analyze a space-time $p$-adaptive discontinuous Galerkin method for nonlinear acoustics. We first present the underlying mathematical model, which is based on a recently derived formulation involving, in particular, only first order in time derivatives. We then propose a spacetime discontinuous Galerkin discretization of this model, combining a symmetric Friedrichs systems discretization for symmetric hyperbolic systems with an interior penalty discretization for damping terms. The resulting nonlinear system is solved using Newton's method. Next, we present a well-posedness analysis of the discrete problem. The analysis begins with a linearized system, for which stability is shown. Using a fixed point argument, these results are extended to the fully discrete nonlinear system, yielding a priori error estimates in a natural discontinuous Galerkin norm. Finally, we present numerical experiments demonstrating the parallel solvability of the spacetime formulation and the effectiveness of p-adaptivity. The results confirm the theoretical convergence rates and show that adaptive refinement can reduce the number of degrees of freedom required to accurately approximate selected goal functionals. Moreover, the experiments demonstrate that the model reproduces characteristic phenomena of nonlinear acoustics, such as harmonic generation, thereby validating the proposed model.