A Modal-Decay-Based Shock-Capturing Approach for High-Order Flux Reconstruction Method

The increasing demand for high-fidelity simulations of compressible turbulence on complex geometries poses a number of challenges for numerical schemes, and plenty of high-order methods have been developed. The high-order methods may encounter spurious oscillations or even blow up for strongly compressible flows, and a number of approaches have been developed, such as slope limiters and artificial viscosity models. In the family of artificial viscosity, which measures smoothness using the modal coefficients, the averaged modal decay (MDA) model employs all of the modes instead of only the highest mode as in the highest modal decay (MDH) model, which tends to underestimate the smoothness. However, the MDA approach requires high-order accuracy (usually (Formula presented.) ) to deliver a reliable estimation of smoothness. In this work, an approach used to extend the MDA model to lower orders, such as (Formula presented.) and (Formula presented.), referred to as MDAEX, was proposed, where neighboring elements were incorporated to involve more information in the estimation process. A further controlling of the value of artificial viscosity was also introduced. The proposed model was applied to several typical benchmark cases and compared with other typical models. The results show that the MDAEX model recovers the expected accuracy better than the MDA model for (Formula presented.) and (Formula presented.) and captures flow structures well for shock-dominated flows.