For the general class of residual distribution (RD) schemes, including many finite element (such as continuous/discontinuous Galerkin) and flux reconstruction methods, an approach to construct entropy conservative/ dissipative semidiscretizations by adding suitable correction terms has been proposed by Abgrall (J.~Comp.~Phys. 372: pp. 640--666, 2018). In this work, the correction terms are characterized as solutions of certain optimization problems and are adapted to the SBP-SAT framework, focusing on discontinuous Galerkin methods. Novel generalizations to entropy inequalities, multiple constraints, and kinetic energy preservation for the Euler equations are developed and tested in numerical experiments. For all of these optimization problems, explicit solutions are provided. Additionally, the correction approach is applied for the first time to obtain a fully discrete entropy conservative/dissipative RD scheme. Here, the application of the deferred correction (DeC) method for the time integration is essential. This paper can be seen as describing a systematic method to construct structure preserving discretization, at least for the considered example.

翻译：Abgrall(J. ~Comp.~Phys. 372: pp. 640-666, 2018年)建议,对于一般的剩余分配(RD)计划类别,包括许多有限要素(如连续/不连续的Galerkin)和通量重建方法,采用一种方法,通过添加适当的修正术语来构建对流保守/消散半分化的微粒子保守/消散半分化(J. ~Comp. ~Phys. 372: pp. 640-666, 2018年),在这项工作中,修正术语被定性为某些优化问题的解决办法,并适应SBP-SAT框架,侧重于不连续的Galerkin方法。本文可以视为描述一种系统化结构维护离散化的方法,至少对于深思熟虑的示例而言。