We present a novel quasi-conservative arbitrary high order accurate ADER (Arbitrary-Derivative) discontinuous Galerkin method allowing to efficiently use a non-conservative form of the considered partial differential system, so that the governing equations can be solved directly in the most physically relevant set of variables. This is particularly interesting for multi-material flows with moving interfaces and steep, large magnitude contact discontinuities, as well as in presence of highly non-linear thermodynamics. However, the non-conservative formulation of course introduces a conservation error which would normally lead to a wrong approximation of shock waves. Hence, from the theoretical point of view, we give a formal definition of the conservation defect of non-conservative schemes and we analyze this defect providing a local quasi-conservation condition, which allows us to prove a . Within this formalism, we also reformulate classical results concerning smooth solutions, contact discontinuities and moving interfaces. Then, to deal with shock waves in practice, we exploit the framework of the so-called subcell finite volume (FV) limiter, so that, in troubled cells appropriately detected, we can incorporate a local conservation correction. Our corrected FV update entirely removes the local conservation defect, allowing, at least formally, to fit in the hypotheses of the proposed modified Lax–Wendroff theorem. Here, the shock-triggered troubled cells are detected by combining physical admissibility criteria, a discrete maximum principle and a shock sensor inspired by Lagrangian hydrodynamics.

中文翻译：

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非保守变量双曲系统的不连续伽辽金方案：具有子单元有限体积修正的准保守公式


我们提出了一种新颖的准保守任意高阶精确 ADER（任意导数）不连续伽辽金方法，允许有效地使用所考虑的偏微分系统的非保守形式，以便可以直接以最物理相关的方式求解控制方程变量集。这对于具有移动界面和陡峭、大规模接触不连续性以及存在高度非线性热力学的多材料流动特别有趣。然而，非保守公式当然会引入守恒误差，这通常会导致冲击波的错误近似。因此，从理论的角度来看，我们给出了非保守方案的守恒缺陷的正式定义，并分析了该缺陷，提供了局部准守恒条件，这使我们能够证明 。在这种形式主义中，我们还重新表述了有关平滑解决方案、接触不连续性和移动界面的经典结果。然后，为了在实践中处理冲击波，我们利用所谓的子单元有限体积（FV）限制器的框架，以便在适当检测到的问题单元中，我们可以结合局部守恒校正。我们修正的 FV 更新完全消除了局部守恒缺陷，至少在形式上允许符合所提出的修改后的 Lax-Wendroff 定理的假设。在这里，通过结合物理可接受标准、离散最大原理和受拉格朗日流体动力学启发的冲击传感器来检测由冲击触发的问题细胞。