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

TL;DR

The authors develop a space-time discontinuous Galerkin method for a nonlinear acoustics model formulated with first-order time derivatives, enabling stable, parallelizable simulations with adaptive refinement. The method integrates a symmetric Friedrichs discretization for the hyperbolic part and an interior-penalty scheme for damping, solving the resulting nonlinear system via Newton iterations. A rigorous well-posedness and a priori error analysis in a natural DG norm is provided for both linearized and nonlinear problems, supported by convergence results that tie errors to projection approximations and data perturbations. Numerical experiments in two dimensions demonstrate parallel solvability, effective $p$-adaptive refinement, and the reproduction of nonlinear phenomena such as harmonic generation, highlighting the practical impact for high-fidelity acoustic simulations.

Abstract

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.

Figures (7)

• Figure 1: DG-norm error and $L^2$ space-time error comparison between known and discrete solutions for different time degrees $p$ and space degrees $q$ in dependence to the total degrees of freedom (DoFs) used. The extrapolated convergence order (ECO) is indicated at the right end of each line.
• Figure 2: Numerical solution for $n_r = 7$ and $p = q = 1$ at different time steps. The approximated pressure $p_h$ is plotted.
• Figure 3: Numerical solution $p_h$ for $n_r = 7$ and $p = q = 1$ on the space-time domain.
• Figure 4: At the top, the time evolution of the pressure at $(x,y) = (0.246094,\, 0.246094)$ is shown. The blue dotted line corresponds to the pressure component of the solution $u_h^L = (p_h^L, v_h^L)$ obtained with $\gamma = \delta = \eta = \theta = 0$, while the orange solid line represents the pressure from the full nonlinear model. At the bottom, the corresponding magnitudes of the discrete Fourier transforms of $p_h$ and $p_h^L$ are displayed.
• Figure 5: The $L^2$ norm of the nonlinear global error estimator $\tilde{\varsigma}$ is shown in dependence of the degree of freedoms of each adaptive iteration.

Theorems & Definitions (24)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Remark 3.1
• Remark 3.2