Abstract

In this paper a relaxed formulation of the a posteriori Multi-dimensional Optimal Order Detection (MOOD) limiting approach is introduced for weighted essentially non-oscillatory (WENO) finite volume schemes on unstructured meshes. The main goal is to minimise the computational footprint of the MOOD limiting approach by employing WENO schemes—by virtue of requiring a smaller number of cells to reduce their order of accuracy compared to an unlimited scheme. The key characteristic of the present relaxed MOOD formulation is that the Numerical Admissible Detector (NAD) is not uniquely defined for all orders of spatial accuracy, and it is relaxed when reaching a 2nd-order of accuracy. The augmented numerical schemes are applied to the 2D unsteady Euler equations for a multitude of test problems including the 2D vortex evolution, cylindrical explosion, double-Mach reflection, and an implosion. It is observed that in many events, the implemented MOOD paradigm manages to preserve symmetry of the forming structures in simulations, an improvement comparing to the non-MOOD limited counterparts which cannot be easily obtained due to the multi-dimensional reconstruction nature of the schemes.

Highlights

  • Coupled with increasingly larger computational power, modern schemes are mature enough to attack intensive gasdynamic and aerodynamic problems, as well as simulations of relativistic fluids with magnetic fields [1]

  • We present the numerical simulations employed to assess the performance of the weighted essentially non-oscillatory (WENO) schemes with the relaxed Multi-dimensional Optimal Order Detection (MOOD) approach in terms of robustness, accuracy and computational efficiency for the solution of the Euler equations in 2D setups

  • The relaxed MOOD algorithm has a significantly smaller computational footprint compared to the original MOOD approach, and at the same time it recovers the theoretical order of accuracy of the WENO schemes, since there are fewer cells that will drop their solution to 1st-order

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Summary

Introduction

Coupled with increasingly larger computational power, modern schemes are mature enough to attack intensive gasdynamic and aerodynamic problems, as well as simulations of relativistic fluids with magnetic fields [1]. The key benefit of the MOOD approach is that it increases the robustness of the numerical framework employed since the solutions at any cell that do not conform to certain, pre-selected criteria are recomputed with spatially lower-order numerical schemes until said criteria are universally satisfied This solution-checking strategy is performed in an a posteriori fashion once the spatial discretisation and fluxes have been computed and the solution is advanced in time. The UCNS3D [55,56] solver employed in this study by Tsoutsanis et al (2018) [56] provides a robust OpenMP-MPI parallel framework for the multi-dimensional unsplit FV numerical computation This permits the simulation of the late-time mixing of, and the strongly-nonlinear evolution of single- and multi-mode RMIs, and Kelvin–Helmholtz instabilities (KHIs), in all spatial dimensions.

General computational framework
Numerical framework
Spatial discretisation
Applications
Cylindrical explosion
Double Mach reflection
Findings
Conclusions

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