Whenever we deal with conservation laws, uniqueness of weak solutions can be guaranteed by the entropy inequality, which derives from the physical entropy. We study the residual of this inequality, which represents the numerical entropy production by the approximation scheme we are considering. This idea has been introduced and exploited in Runge Kutta finite volume methods, where the numerical entropy production has been used as an indicator in adaptive schemes, since it scales as the local truncation error of the method for smooth solutions and it highlights the presence of discontinuities and their kind. The aim of this work is to extend this idea to ADER timestepping techniques. We show that the numerical entropy production can be defined also in this context and it provides a scalar quantity computable for each spacetime volume which, under grid refinement, decays to zero with the same rate of convergence of the scheme for smooth solutions it's bounded on contact discontinuities and divergent on shock waves. Theoretical results are proven in a multidimensional setting. We also present numerical evidence showing that it is essentially negative definite. Moreover, we propose an adaptive scheme that uses the numerical entropy production as smoothness indicator. The scheme locally modifies its order of convergence with the purpose of removing the oscillations due to the high-order of accuracy of the scheme.

翻译：在处理守恒定律时，弱解的唯一性可由熵不等式保证，该不等式源于物理熵。我们研究此不等式的残差，它代表所考虑近似格式产生的数值熵生成。这一概念已在龙格-库塔有限体积方法中引入并应用，其中数值熵生成被用作自适应格式的指示器——对于光滑解，其缩放比例与方法的局部截断误差一致，并能凸显间断及其类型的特征。本工作的目标是将此思想推广至ADER时间步进技术。我们证明在此框架下同样可以定义数值熵生成，它能为每个时空体积提供可计算的标量值：在网格细化过程中，该值对光滑解以与格式相同的收敛速率衰减至零，在接触间断处保持有界，在激波处呈现发散特性。理论结果在多维条件下得到证明。我们还提供了数值证据，表明其本质上具有负定性。此外，我们提出一种采用数值熵生成作为光滑度指示器的自适应格式。该格式通过局部调整收敛阶数，旨在消除由高阶精度格式引起的振荡现象。