A high‐order Runge‐Kutta discontinuous Galerkin method with a subcell limiter on adaptive unstructured grids for two‐dimensional compressible inviscid flows

Abstract

Summary A robust, adaptive unstructured mesh refinement strategy for high‐order Runge‐Kutta discontinuous Galerkin method is proposed. The present work mainly focuses on accurate capturing of sharp gradient flow features like strong shocks in the simulations of two‐dimensional inviscid compressible flows. A posteriori finite volume subcell limiter is employed in the shock‐affected cells to control numerical spurious oscillations. An efficient cell‐by‐cell adaptive mesh refinement is implemented to increase the resolution of our simulations. This strategy enables to capture strong shocks without much numerical dissipation. A wide range of challenging test cases is considered to demonstrate the efficiency of the present adaptive numerical strategy for solving inviscid compressible flow problems having strong shocks.

A Priori Neural Networks Versus A Posteriori MOOD Loop: A High Accurate 1D FV Scheme Testing Bed

“…A perceptron of layer , δ ,j , receives as input data the signals y −1 from the perceptrons located in the previous layer. From these data using the linear and non-linear operations (17) and (18), the perceptron produces a single output signal y ,p , that is…”

“…Indeed this a posteriori approach detects if the unlimited candidate solution (at the next time-step) fulfills some validity criteria: computer (NaN), physical (positivity preserving) and numerical admissibility (essentially non-oscillatory behavior based on relaxed Discrete Maximum Principle (DMP)). This stage is in fact a troubled cell detector [25,35,15,18]. Then for any troubled cell, the numerical solution is discarded and locally recomputed starting again at the beginning of the time-step with a more robust and less accurate scheme.…”

A Compact Subcell WENO Limiting Strategy Using Immediate Neighbors for Runge-Kutta Discontinuous Galerkin Methods for Unstructured Meshes

“…These new cells are used for the WENO reconstruction. This formulation is different from the subcell limiting strategy of Dumbser et al [9,10] and Giri et al [13] which is much more accurate but requires more effort as they use subcells in an a posteriori limiting strategy. This limiting strategy can be used with any of the WENO reconstructions given in [15] (called type I WENO reconstruction), or [5] (called type II WENO reconstruction) or [16] (mixed reconstruction) or the more recent methods given in [17] and [2] by dividing the immediate neighbors appropriately.…”

“…This limiter was successfully applied to turbulent shallow water equations by Busto et al in [12]. This subcell limiter is further refined using an Adaptive Mesh Refinement (AMR) technique by Giri and Qiu in [13]. Sonntag and Munz in [14] developed a subcell limiter which uses finite volume subcells, where each subcell is associated with one degree of freedom within the DG grid cell and parallelized it effectively.…”

A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

“…It is important to note that these choices involving the element type, basis functions, and approximation of inner products all have a major impact on the performance of the resulting DG scheme in terms of computational complexity and robustness due, e.g., to the presence of spurious oscillations near discontinuities that result in unphysical solution states (like negative density or pressure) or aliasing instabilities. Many mechanisms exist in the DG community to combat spurious oscillations (i.e., shock capturing) such as slope [3,5,35] or WENO [36,37] limiters, filtering [29,38,39], finite volume sub-cells [40][41][42], MOOD-type limiting [43][44][45][46][47], or artificial viscosity [48,49]. The issue of shock capturing will not be discussed further.…”