A review of discontinuous Galerkin time-stepping methods for wave propagation problems

TL;DR

The chapter analyzes high-order discontinuous Galerkin time-stepping schemes for wave-type problems arising from space discretization, detailing two formulations: dG2 for second-order systems and dG1 for first-order systems. It develops a unified variational framework, proves well-posedness and energy-based stability, and provides a priori error estimates in corresponding energy seminorms. Algebraic formulations per time slab yield efficient, structured linear systems, with dG1 employing a Schur-complement approach that can improve conditioning. Numerical experiments across acoustic, poroelastic, and coupled poroelastic-acoustic scenarios confirm the theoretical rates and demonstrate that dG1 offers better accuracy and conditioning than dG2, guiding practical implementation choices in complex wave simulations.

Abstract

This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave propagation problems. Two formulations, tailored for second- and first-order systems of ordinary differential equations, are discussed within a generalized framework, assessing their stability, accuracy, and computational efficiency. Theoretical results are supported by various illustrative examples that validate the findings, enhancing the understanding and applicability of these methods in practical scenarios.

Figures (10)

• Figure 1: Time domain partition: values $t_n^+$ and $t_n^-$ are also reported.
• Figure 2: Polygonal meshes for the test case of Section \ref{['sec::acoustic_example']} (left) and Section \ref{['sec::poro_example']} (right).
• Figure 3: Test case of Section \ref{['sec::acoustic_example']}. Computed errors $|e_\varphi|_{{\mathcal{A}}}$ for the dG2 (left), and $|(e_\varphi,e_\psi)|_{{\mathcal{B}}}$ for the dG1 (right) in logarithmic scale as a function of the time step $\Delta t$ for different polynomial degrees $r=1,2,3,4$. We set $N_{el}=400$ polygonal elements and a space polynomial degree equal to $7$.
• Figure 4: Test case of Section \ref{['sec::acoustic_example']}. Computed errors $\|e_\varphi\|_{0}$ for the dG2 (left) and dG1 (right) method in logarithmic scale as a function of the time step $\Delta t$ for different polynomial degrees $r=1,2,3,4$. We set $N_{el}=400$ polygonal elements and a space polynomial degree equal to $7$.
• Figure 5: Test case of Section \ref{['sec::acoustic_example']}. Computed errors for the dG2 (left) and dG1 (right) method in semilogarithmic scale as a function of the polynomial degree $r$ for $\Delta t = 0.02$. We use $N_{el}=400$ polygonal elements and a space polynomial degree equal to $7$.

Theorems & Definitions (17)

• Theorem 1
• proof
• Remark 1
• Theorem 2: Well posedness of \ref{['eq::wf_dgII']}
• Proposition 1: Stability of dG2
• proof
• Theorem 3: Convergence of dG2
• proof
• Remark 2
• Proposition 2: Stability of dG1