We present a novel quasi-conservative arbitrary high order accurate ADER discontinuous Galerkin (DG) 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 modified Lax-Wendroff theorem. Then, to deal with shock waves in practice, we exploit the framework of the so-called a posteriori 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. To prove the capabilities of our novel approach, first, we show that we are able to recover the same results given by conservative schemes on classical benchmarks for the single-fluid Euler equations. We then conclude by showing the improved reliability of our scheme on the multi-fluid Euler system on examples like the interaction of a shock with a helium bubble.

翻译：本文提出了一种新颖的拟守恒任意高阶精度ADER间断伽辽金方法，该方法允许高效地使用偏微分方程组的非守恒形式，从而能够直接在最具有物理相关性的变量集中求解控制方程。这对于具有运动界面、陡峭大强度接触间断的多材料流动，以及存在高度非线性热力学的情况尤其重要。然而，非守恒形式自然会引入守恒误差，这通常会导致对激波的不正确近似。因此，从理论角度，我们给出了非守恒格式守恒缺陷的形式化定义，并通过分析该缺陷提出了局部拟守恒条件，从而得以证明一个修正的Lax-Wendroff定理。随后，为了在实际中处理激波，我们利用所谓的后验子单元有限体积限制器框架，使得在适当检测到的"问题单元"中，可以引入局部守恒修正。我们修正后的有限体积更新完全消除了局部守恒缺陷，从而至少在形式上符合所提出的修正Lax-Wendroff定理的假设条件。在此，通过结合物理可容性准则、离散极大值原理以及受拉格朗日流体力学启发的激波探测器，来识别由激波触发的问题单元。为证明新方法的能力，我们首先展示了在单流体欧拉方程的经典基准测试中能够复现守恒格式给出的相同结果。最后，通过在诸如激波与氦气泡相互作用等算例上展示多流体欧拉系统中本格式的更高可靠性，完成了全文的论证。