"A posteriori" limited high order and robust schemes for transient simulations of fluid flows in gas dynamics

In this paper, we propose a novel finite element-type Residual Distribution scheme for time-dependent hyperbolic systems of conservation laws for the Euler system of gas dynamics for strong interacting discontinuities. The goal of our method is to allow for high-order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behaviour in the regions of steep gradients, in particular across shocks. Following the Multidimensional Optimal Order Detection (MOOD) (Clain et al., Journal of Computational Physics 2011; Diot et al., Computer & Fluids 2012) approach, a candidate solution is computed at a next time level via a high-order accurate residual distribution scheme (Abgrall, Journal of Scientific Computing 2017; Abgrall et al., Computers & Mathematics with Applications 2018). A so-called detector determines if the candidate solution reveals any spurious oscillation or numerical issue and, if so, only the troubled cells are locally recomputed via a more dissipative scheme. This allows to design a family of "a posteriori" limited, robust and positivity preserving, as well as high accurate, non-oscillatory and effective residual distribution schemes. Among the detecting criteria of the novel MOOD strategy, two different approaches from literature, based on the work of Clain et al. (Journal of Computational Physics 2011), Diot et al. (Computer & Fluids 2012) and on Vilar (Journal of Computational Physics 2018), are investigated. Numerical examples in 1D and 2D, on structured and unstructured meshes, are proposed to assess the effective order of accuracy for smooth flows, the non-oscillatory behaviour on shocked flows, the robustness and positivity preservation on more extreme flows.

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