Lifting solutions of quasilinear 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...

H-matrix Preconditioners in Convection-Dominated Problems

Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit H-matrix techniques to approximate the LU -decompositions of stiffness matrices as they appear in (finite element or finite difference) discretizations of convectiondominated elliptic partial differential equations. These sparse H-matrix approximations may then be used as preconditi...

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