The XFEM for high-gradient solutions in convection-dominated problems

Lifting solutions of quasilinear convection-dominated problems

The steady state of the quasilinear convection-diffusion-reaction equation ut −∇(D(u)∇u) + b(u)∇u+ c(u) = 0 (1) is studied. Depending on the ratio between convection and diffusion coefficients, equation (1) ranges from parabolic to almost hyperbolic. From a numerical point of view two main difficulties can arise related with the existence of layers and/or the non smoothness of the coefficients....

Discontinuous Galerkin Methods for Convection-Dominated Problems

In this paper, we review the development of the Runge–Kutta discontinuous Galerkin (RKDG) methods for non-linear convection-dominated problems. These robust and accurate methods have made their way into the main stream of computational fluid dynamics and are quickly finding use in a wide variety of applications. They combine a special class of Runge–Kutta time discretizations, that allows the m...

Hierarchical Matrices for Convection-Dominated Problems

Hierarchical matrices provide a technique to efficiently compute and store explicit approximations to the inverses of stiffness matrices computed in the discretization of partial differential equations. In a previous paper, Le Borne [2003], it was shown how standard H-matrices must be modified in order to obtain good approximations in the case of a convection dominant equation with a constant c...

Numerical Methods for Scalar Convection-Dominated Problems

Genuinely Nonlinear Models for Convection-dominated Problems

This paper introduces a general, nonlinear subgrid-scale (SGS) model, having bounded artiicial viscosity, for the numerical simulation of convection-dominated problems. We also present a numerical comparison (error analysis and numerical experiments) between this model and the most common SGS model of Smagorinsky, which uses a p-Laplacian regularization. The numerical experiments for the 2-D co...