Section: New Results

High order discretizations on unstructured meshes

• Participants: Héloise Beaugendre, Cécile Dobrzynski, Léo Nouveau, Mario Ricchiuto, Quentin Viville

• Corresponding member: Héloise Beaugendre

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 technique 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 [122]. 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 [16]. This approach has been combined with $r-$adaptation techniques to account for moving bodies and validated on simulations involving flapping wings, and computations of ices block trajectories in the framework of the STORM project [56], [46];

• 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. This year, we extended the p-adaptation method to Navier-Stokes equations and coupled it with immersed boundary method.

• We have studied the high order approximation of problems with dispersion and suggested a route allowing to construct high order methods (up to order 4) allowing to obtain the same accuracy for the solution, and for its first and second order derivatives. Initial validation for the approach proposed has been shown for the time dependent KdV equations [14], [49].