Central-Upwind Schemes for Shallow Water Models.
Dienstag, 12.7.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
I will first give a brief review on simple and robust central-upwind\nschemes for hyperbolic conservation laws.\n\nI will then discuss their application to the Saint-Venant system of\nshallow water equations. This can be done in a straightforward manner, but\nthen the resulting scheme may suffer from the lack of balance between the\nfluxes and (possibly singular) geometric source term, which may lead to a\nso-called numerical storm, and from appearance of negative values of the\nwater height, which may destroy the entire computed solution. To\ncircumvent these difficulties, we have developed a special technique,\nwhich guarantees that the designed second-order central-upwind scheme is\nboth well-balanced and positivity preserving.\n\nFinally, I will show how the scheme can be extended to the two-layer\nshallow water equations and to the Savage-Hutter type model of submarine\nlandslides and generated tsunami waves, which, in addition to the\ngeometric source term, contain nonconservative interlayer exchange terms.\nIt is well-known that such terms, which arise in many different multiphase\nmodels, are extremely sensitive to a particular choice their numerical\ndiscretization. To circumvent this difficulty, we rewrite the studied\nsystems in a different way so that the nonconservative terms are\nmultiplied by a quantity, which is, in all practically meaningful cases,\nvery small. We then apply the central-upwind scheme to the rewritten\nsystem and demonstrate robustness and superb performance of the proposed\nmethod on a number numerical examples.
Hybrid DG Methods for Incompressible Flow.
Dienstag, 26.7.11, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We propose a class of hybrid discontinuous Galerkin methods for the numerical\n solution of incompressible flow problems. Our approach yields consistent and locally\n conservative discretizations, and allows to treat non-conforming h and p-refinements\n in a natural way.\n\nBy explicit construction of a Fortin-operator, we show that the proposed methods\n are inf-sup stable with a constant which is independent of the meshsize, and only\n slightly dependent on the polynomial degree. This allows to derive a-priori error\n estimates for the Stokes problem, which are optimal in the mesh size and only\n slightly sub-optimal in the polynomial degree.\n\nWe also discuss the extension to stationary Oseen and Navier-Stokes equations, and illustrate our theoretical results with numerical tests.\n\n