A robust and adaptive recovery‐based discontinuous Galerkin method for the numerical solution of convection–diffusion equations

Abstract

SUMMARYIn this paper, we introduce and test the enhanced stability recovery (ESR) scheme. It is a robust and compact approach to the computation of diffusive fluxes in the framework of discontinuous Galerkin methods. The scheme is characterized by a new recovery basis and a new procedure for the weak imposition of Dirichlet boundary conditions. These features make the method flexible and robust, even in the presence of highly distorted meshes. The implementation is simplified with respect to the original recovery scheme (RDG1x). Furthermore, thanks to the proposed approach, a robust implementation of p‐adaptive algorithms is possible. Numerical tests on unstructured grids show a convergence rate equal to p + 1, where p is the reconstruction order. Comparisons are shown with the original recovery scheme RDG1x and the widely used BR2 method. Results show a significantly larger stability region for the proposed discretization when explicit Runge–Kutta time integration is employed. Interestingly, this advantage grows quickly when the reconstruction order is increased. The proposed procedure for the weak imposition of Dirichlet boundary conditions does not need the introduction of ghost cells, and it is truly local because it does not require data exchange with other elements. It can be easily used with curvilinear wall elements. Several test cases are considered. They include some benchmark tests with the heat equation and compressible Navier–Stokes equations, with test cases designed also to evaluate the behaviour of the scheme with very stretched elements and separated flows. Copyright © 2014 John Wiley & Sons, Ltd.

Progress Towards the Application of the Recovery-Based Disontinuous Galerkin Method to Practical Flow Physics Problems

“…The details of the method, further developed by Lo,3 are given in the next section. The basic philosophy of the method has been applied in the formation of two new algorithms that we are aware of, namely Borrel & Ryan's Elastoplast DG method 11 and the EnhancedStability Recovery (ESR) scheme of Ferrero et al, 12 but this paper will focus on the evolved forms of RDG developed by Lo, which preserve extremely high order of accuracy.…”

Control of a Supersonic Inlet in Off-Design Conditions with Plasma Actuators and Bleed

“…The governing equations are discretized by means of an unstructured solver which is based on the method of lines: a finite volume discretization is adopted in space while time integration is performed by means of the linearized implicit Euler scheme. The solver, which can be use in both finite volume or discontinuous Galerkin mode, has been verified and validated on compressible inviscid flows [42,43], laminar flows [44] and turbulent flows [45][46][47][48][49][50].…”

Global and local POD models for the prediction of compressible flows with DG methods

“…In this work the convective part of the numerical flux is computed by solving a Riemann problem with the method proposed by Osher and Solomon [37] and implemented according to Pandolfi [38]. The diffusive part of the numerical flux is computed by means of a recovery based approach [26]. The volume and surface integrals in Eq.…”

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