Siva Nadarajah

Samuel Quaegebeur, Siva Nadarajah

The flux reconstruction (FR) method has gained popularity within the research community. The approach has been demonstrated to recover high-order methods such as the discontinuous Galerkin (DG) method. Stability analyses have been conducted for a linear advection problem leading to the energy stable flux reconstruction (ESFR) methods also named Vincent-Castonguay-Jameson-Huynh (VCJH) methods. ESFR schemes can be viewed as DG schemes with modally filtered correction fields. Using this class of methods, the linear advection diffusion problem has been shown to be stable using the local discontinuous Galerkin scheme (LDG) to compute the viscous numerical flux. This stability proof has been extended for linear triangular and tetrahedra elements. Although the LDG scheme is commonly used, it requires, on particular meshes, a wide stencil, which raises the computational cost. As a consequence, many prefer the compact interior penalty (IP) or the Bassi and Rebay II (BR2) numerical fluxes. This article, for the first time, derives, for both schemes, a condition on the penalty term to ensure stability. Moreover the article establishes that for both the IP and BR2 numerical fluxes, the stability of the ESFR scheme is independent of the auxiliary correction field. A von Neumann analysis is conducted to study the maximal time step of various ESFR methods.

Philip Zwanenburg, Siva Nadarajah

We provide numerical evidence demonstrating the necessity of employing a superparametric geometry representation in order to obtain optimal convergence orders on two-dimensional domains with curved boundaries when solving the Euler equations using Discontinuous Galerkin methods. However, concerning the obtention of optimal convergence orders for the Navier-Stokes equations, we demonstrate numerically that the use of isoparametric geometry representation is sufficient for the case considered here.

Samuel Quaegebeur, Siva Nadarajah, Farshad Navah et al.

Recovering some prominent high-order approaches such as the discontinuous Galerkin (DG) or the spectral difference (SD) methods, the flux reconstruction (FR) approach has been adopted by many individuals in the research community and is now commonly used to solve problems on unstructured grids over complex geometries. This approach relies on the use of correction functions to obtain a differential form for the discrete problem. A class of correction functions, named energy stable flux reconstruction (ESFR) functions, has been proven stable for the linear advection problem. This proof has then been extended for the diffusion equation using the local discontinuous Galerkin (LDG) scheme to compute the numerical fluxes. Although the LDG scheme is commonly used, many prefer the interior penalty (IP), as well as the Bassi and Rebay II (BR2) schemes. Similarly to the LDG proof, this article provides a stability analysis for the IP and the BR2 numerical fluxes. In fact, we obtain a theoretical condition on the penalty term to ensure stability. This result is then verified through numerical simulations. To complete this study, a von Neumann analysis is conducted to provide a combination of parameters producing the maximal time step while converging at the correct order. All things considered, this article has for purpose to provide the community with a stability condition while using the IP and the BR2 schemes.

Carolyn M. V. Pethrick, Siva Nadarajah

We present a high-order space-time discretization equipped with fully-discrete entropy stability properties for general choices of volume and surface quadrature rules. The formulation uses flux reconstruction (FR) in the spatial dimension paired with a discontinuous Galerkin (DG) method in the temporal dimension. The result is a fully-implicit system using polynomial bases in space and time. An energy-stable discretization is applied to the linear advection equation, yielding optimal $p+1$ convergence for small FR correction parameters and $p$ convergence at the same filter strength as method-of-lines implementations. We can thus recover the space-time equivalent to schemes such as DG, Huynh's FR, or spectral difference through a single parameter $c$. We follow with a similar space-time nonlinearly-stable flux reconstruction (ST-NSFR) scheme, which uses skew-symmetric stiffness operators in both space and time. The ST-NSFR scheme is fully-discretely entropy preserving using the $c_{DG}$ parameter or entropy-stable for small $c$. Numerical experiments using the linear advection and Euler equations confirm convergence orders and stability properties. The advantage of FR in a space-time context is demonstrated by a reduction in computational cost up to about $70\%$ as $c$ is increased.