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search.filters.applied.f.access: openAccess × Author: John, Volker × Author: Novo, Julia × Type: Text × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × Subject: WENO schemes ×

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    On (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) John, Volker; Novo, Julia
    Finite element and finite difference discretizations for evolutionary convection-diffusion-reaction equations in two and three dimensions are studied which give solutions without or with small under- and overshoots. The studied methods include a linear and a nonlinear FEM-FCT scheme, simple upwinding, an ENO scheme of order 3, and a fifth order WENO scheme. Both finite element methods are combined with the Crank--Nicolson scheme and the finite difference discretizations are coupled with explicit total variation diminishing Runge--Kutta methods. An assessment of the methods with respect to accuracy, size of under- and overshoots, and efficiency is presented, in the situation of a domain which is a tensor product of intervals and of uniform grids in time and space. Some comments to the aspects of adaptivity and more complicated domains are given. The obtained results lead to recommendations concerning the use of the methods.
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    Projection methods for incompressible flow problems with WENO finite difference schemes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Frutos, Javier de; John, Volker; Novo, Julia
    Weighted essentially non-oscillatory (WENO) finite difference schemes have been recommended in a competitive study of discretizations for scalar evolutionary convectiondiffusion equations [?]. This paper explores the applicability of these schemes for the simulation of incompressible flows. To this end, WENO schemes are used in several nonincremental and incremental projection methods for the incompressible Navier-Stokes equations. Velocity and pressure are discretized on the same grid. A pressure stabilization Petrov-Galerkin (PSPG) type of stabilization is introduced in the incremental schemes to account for the violation of the discrete inf-sup condition. Algorithmic aspects of the proposed schemes are discussed. The schemes are studied on several examples with different features. It is shown that the WENO finite difference idea can be transferred to the simulation of incompressible flows. Some shortcomings of the methods, which are due to the splitting in projection schemes, become also obvious.