Section: New Results

High order discretizations on unstructured meshes

Participants : Héloise Beaugendre [Corresponding member] , Cécile Dobrzynski, Léo Nouveau, Mario Ricchiuto, Quentin Viville.

Our work on high order unstructured discretizations this year has pursued three main avenues:

• We have extended the team's previous work on the consistent residual based approximation of viscous flow equations to the framework of Immersed Boundary Methods (IBM). This is an increasingly popular approach in Computational Fluid Dynamics as it simplifies the mesh generation problem. In our work, we consider a echnique based on the addition of a penalty term to the Navier-Stokes equations to account for the wall boundary conditions. To adapt the residual distribution method method to the IBM, we developed a new formulation based on a Strang splitting appproach in time. This approach, couples in a fully consistent manner an implicit asymptoticly exact integration procedure of the penalization ODE, with the explicit residual distribution discretization for the Navier-Stokes equations, based on the method proposed in (Ricchiuto and Abgrall, J.Comput.Phys 229, 2010). The ODE integrator provides an operator which is exact up to orders ${\eta }^2$, with $\eta$ the penalty parameter assuming values of the order of ${10}^{-10}$. To preserve the accuracy of the spatial discretization in the Navier-Stokes step, we have introduced, in vicinity of the penalised region, a modification of the solution gradient reconstruction required for the evaluation of the viscous fluxes. We have shown formally and numerically that the approach proposed is second order accurate for smooth solutions, and shown its potential when combined with unstructured mesh adaptation strategies w.r.t. the (implicitly described) solid walls. This work has been accepted on Comp.Meth.Appl.Mech.Eng. ;

• Another research axis consists in proposing a novel approach that allows to use p-adaptation with continuous finite elements. Under certain conditions, primarily the use of a residual distribution scheme, it is possible to avoid the continuity constraint imposed to the approximate solution, while still retaining the advantages of a method using continuous finite elements. The theoretical material, the complete numerical method and practical results show as a proof of concept that p-adaptation is possible with continuous finite elements. Its extension to penalized Navier-Stokes equations are under progress ;

• We have continued the study of the properties of residual based methods in the time dependent case. We have been able to further characterize one of the variants of the approach proposed in (Ricchiuto and Abgrall, J.Comput.Phys 229, 2010) in terms of preservation of the positivity of the density showing this property in practical applications involving the shallow water equations [130] . The impact of the simplified construction leading to these schemes has also been investigated. In particular, we have shown that despite the additional complexity associated to the inversion of the mass matrix, non-linear methods providing monotone solution and yet featuring linear mass matrices can be constructed [142] . These methods, have been shown to have some potential w.r.t. fully diagonal approaches as those used in (Ricchiuto and Abgrall, J.Comput.Phys 229, 2010), in terms of error as fucntion of CPU time : the non-diagonal schemes showing error reductions up to one order of magnitude. Current work is devoted to the use of other multistage (defect correction type) and multistep (extrapolated methods) techniques, comparing them to space time approaches.