The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between the space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and a high-order time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical diffusion and dispersion. The analysis of these artifacts is well known for finite volume schemes, but it becomes more complex in the DG case. In particular, as far as we know, no analysis of this type has been considered for implicit integration with DG space discretization. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta scheme impacts deeply on the quality of the solution. We analyze dispersion-diffusion properties to select the best combination of the space-time discretization for high Courant numbers. In the second part of this work, we apply our findings to the integration of stiff hyperbolic systems with DG schemes. Implicit time-integration schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements, thereby bypassing the need for minute time-steps. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity of these schemes, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. This approach follows the methodology proposed by Puppo et al. (Commun. Comput. Phys., 2024) for high-order finite volume schemes. Numerical experiments explore the performance of this technique on scalar equations and systems.

翻译：间断伽辽金（DG）格式在双曲守恒律系统中的应用，需要空间离散（通过局部多项式与单元间数值通量实现）与高阶时间积分之间的精细配合，以得到最终更新。一个关键问题在于格式如何通过数值耗散与色散的概念修正解。这些人工效应的分析在有限体积格式中已广为人知，但在DG情形下则更为复杂。特别是，据我们所知，目前尚未有研究针对采用DG空间离散的隐式积分进行此类分析。本工作的第一部分旨在填补这一空白，表明隐式龙格-库塔格式的选择会深刻影响解的质量。我们通过分析色散-耗散特性，为高库朗数情形选取最优的时空离散组合。在第二部分中，我们将所得结论应用于采用DG格式的刚性双曲系统积分。隐式时间积分格式凭借其优越的稳定性，允许仅基于精度要求选择时间步长，从而避免了使用极小时步的必要性。高阶格式需要引入局部空间限制器，这使得整个隐式格式高度非线性。为降低这些格式的数值复杂度，我们提出使用可在解的一阶预测值上预先计算的空间限制器。该方法遵循了Puppo等人（Commun. Comput. Phys., 2024）针对高阶有限体积格式提出的方法论。数值实验探究了该技术在标量方程及系统上的性能表现。