Section: New Results

High order discretizations on unstructured meshes

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

• Corresponding member: Héloise Beaugendre

A $p$-adaptive continuous residual distribution scheme has been proposed. Under certain conditions, primarily the expression of the total residual on a given element K into residuals on the sub-elements of K and the use of a suitable combination of quadrature formulas, it is possible to change locally the degree of the polynomial approximation of the solution. The discrete solution can then be considered non continuous across the interface of elements of different orders, while the numerical scheme still verifies the hypothesis of the discrete Lax–Wendroff theorem which ensures its convergence to a correct weak solution. The construction of our p-adaptive method has been done in the frame of a continuous residual distribution (RD) scheme. Different test cases for non-linear equations at different flow velocities demonstrate numerically the validity of the theoretical results.

As an evolution, a $hp$-adaptive RD scheme for the penalized Navier-Stokes equations has also been developed. The method combines $hp$-adaptation and penalization within a Residual Distribution scheme. The proposed method is an embedded boundary method that provides a simple and accurate treatment of the wall boundary conditions by the technique of penalization and anisotropic mesh adaptation. This method extends the IBM-LS-AUM method to higher order elements and is based on the construction of a $p$-adaptive RD scheme combined with an anisotropic mesh adaptation method. It has been applied to the resolution of the penalized Navier-Stokes equations. The robustness of the method is showed in practice with numerical experiments for different Mach regimes and Reynolds numbers in dimension two and three.

A novel formulation of residual distributions schemes as well as finite volume schemes on unstructured triangulations in curvilinear coordinates has been proposed within the PhD of L. Arpaia [31], generalising the work of [3]. The simulations reveal that the RD method proposed has very low dissipation when compared to the results existing in literature. This is very encouraging for applications in meteorology. A positivity preserving variant of the method has been used to compute large scale impact and inundation of the 2011 Tohoku tsunami [33], [32].