Section: New Results

Adaptation techniques

• Participants: Nicolas Barral, Héloïse Beaugendre, Luca Cirrottola, Algiane Froehly, Mario Ricchiuto.

• Corresponding member: Nicolas Barral

  In [14] we presented an algorithm to perform PDE-based r-adaptation in three-dimensional numerical simulations of unsteady compressible flows on unstructured meshes. A Laplacian-based model for the moving mesh is used to follow the evolving shock-wave patterns in the fluid flow, while the finite volume ALE formulation of the flow solver is employed to implicitly perform a conservative re-mapping of the solution from the previous to the current mesh, at each time step of the simulation.We show the application of this method to compressible flows on three-dimensional geometries. To this aim, an improved relaxation scheme has been developed in order to preserve the validity of the mesh throughout the time simulation in three dimensions, where the geometrical constraints typically restrict the allowable mesh motion.

  Similar adaptive strategies have been investigate for shallow water flows in the context of space-time residual distribution methods [7]. In the scalar case, these schemes can be designed to be unconditionally (w.r.t. the time step) positive, even on the distorted space-time prisms which arise from moving the nodes of an unstructured triangular mesh. Consequently, a local increase in mesh resolution does not impose a more restrictive stability constraint on the time-step, which can instead be chosen according to accuracy or physical requirements. Moreover, schemes of this type are analogous to conservative ALE formulations and automatically satisfy a discrete geometric conservation law. For shallow water flows over variable bed topography, the so-called C-property (retention of hydrostatic balance between flux and source terms, required to maintain the steady state of still, flat, water) can also be satisfied by considering the mass balance equation in terms of free surface level instead of water depth, even when the mesh is moved. Combined with a simple implementation of Laplacian based r-adaptation, this technique has been shown to allow up to 60% CPU time savings for a given error.

  We also extended these methods to curvilinear coordinates to do shallow water simulations on the sphere for oceanographic applications [2]. To provide enhanced resolution of moving fronts present in the flow we consider adaptive discrete approximations on moving triangulations of the sphere. To this end, we re- state all Arbitrary Lagrangian Eulerian (ALE) transport formulas, as well as the volume transformation laws, for a 2D manifold. Using these results, we write the set of ALE-SWEs on the sphere. We then propose a Residual Distribution discrete approximation of the governing equations. Classical properties as the DGCL and the C-property (well balancedness) are reformulated in this more general context. An adaptive mesh movement strategy is proposed. The discrete framework obtained is thoroughly tested on standard benchmarks in large scale oceanography to prove their potential as well as the advantage brought by the adaptive mesh movement.