Numeerisen analyysin ja laskennallisen tieteen seminaari

27.2.2006  klo 14.15  U356

Dmitri Kuzmin, Institute of Applied Mathematics, University of Dortmund

Algebraic constraints and flux limiters for the design of
positivity-preserving finite element schemes

 An algebraic approach to the design of high-resolution finite  element schemes for convection-dominated flow problems is introduced. It is explained how to get rid of nonphysical oscillations and to remove excessive artificial diffusion in regions where the solution is sufficiently smooth. To this end, the discrete operators resulting from a standard Galerkin discretization of the troublesome convective terms are modified so as to enforce the discrete maximum principle without violating mass conservation. A family of algebraic flux correction schemes is derived on the basis of a node-oriented limiting strategy which traces its origins to the multidimensional flux-corrected transport (FCT) algorithm. The algebraic constraints to be imposed are discussed in detail. A new definition of upper and lower bounds leads to a general-purpose flux limiter which guarantees an optimal treatment of stationary and time-dependent problems alike. The use of flux/slope limiters as error indicators for adaptive mesh refinement is promoted. Numerical results are presented for scalar conservation laws as well as for the Euler and Navier-Stokes equations in two and three dimensions.