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- ItemA review of variational multiscale methods for the simulation of turbulent incompressible flows(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Ahmed, Naveed; Rebollo, Tomás Chacón; John, Volker; Rubino, SamueleVarious realizations of variational multiscale (VMS) methods for simulating turbulent incompressible flows have been proposed in the past fifteen years. All of these realizations obey the basic principles of VMS methods: They are based on the variational formulation of the incompressible Navier-Stokes equations and the scale separation is defined by projections. However, apart from these common basic features, the various VMS methods look quite different. In this review, the derivation of the different VMS methods is presented in some detail and their relation among each other and also to other discretizations is discussed. Another emphasis consists in giving an overview about known results from the numerical analysis of the VMS methods. A few results are presented in detail to highlight the used mathematical tools. Furthermore, the literature presenting numerical studies with the VMS methods is surveyed and the obtained results are summarized.
- ItemProjection 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, JuliaWeighted 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.
- ItemSUPG reduced order models for convection-dominated convection-diffusion-reaction equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Iliescu, Traian; John, Volker; Schyschlowa, Swetlana; Wells, DavidThis paper presents a Streamline-Upwind Petrov--Galerkin (SUPG) reduced order model (ROM) based on Proper Orthogonal Decomposition (POD). This ROM is investigated theoretically and numerically for convection-dominated convection-diffusion-reaction equations. The SUPG finite element method was used on realistic meshes for computing the snapshots, leading to some noise in the POD data. Numerical analysis is used to propose the scaling of the stabilization parameter for the SUPG-ROM. Two approaches are used: One based on the underlying finite element discretization and the other one based on the POD truncation. The resulting SUPG-ROMs and the standard Galerkin ROM (G-ROM) are studied numerically. For many settings, the results obtained with the SUPG-ROMs are more accurate. Finally, one of the choices for the stabilization parameter is recommended.
- ItemOn basic iteration schemes for nonlinear AFC discretizations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Jha, Abhinav; John, VolkerAlgebraic flux correction (AFC) finite element discretizations of steady-state convection-diffusionreaction equations lead to a nonlinear problem. This paper presents first steps of a systematic study of solvers for these problems. Two basic fixed point iterations and a formal Newton method are considered. It turns out that the fixed point iterations behave often quite differently. Using a sparse direct solver for the linear problems, one of them exploits the fact that only one matrix factorization is needed to become very efficient in the case of convergence. For the behavior of the formal Newton method, a clear picture is not yet obtained.
- ItemOn (essentially) non-oscillatory discretizations of evolutionary convection-diffusion equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) John, Volker; Novo, JuliaFinite 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.
- ItemSimulations of an ASA flow crystallizer with a coupled stochastic-deterministic approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Bartsch, Clemens; John, Volker; Patterson, Robert I.A.A coupled solver for population balance systems is presented, where the flow, temperature, and concentration equations are solved with finite element methods, and the particle size distribution is simulated with a stochastic simulation algorithm, a so-called kinetic Monte-Carlo method. This novel approach is applied for the simulation of an axisymmetric model of a tubular flow crystallizer. The numerical results are compared with experimental data.
- ItemA local projection stabilization/continuous Galerkin-Petrov method for incompressible flow problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Ahmed, Naveed; John, Volker; Matthies, Gunar; Novo, JuliaThe local projection stabilization (LPS) method in space is considered to approximate the evolutionary Oseen equations. Optimal error bounds independent of the viscosity parameter are obtained in the continuous-in-time case for the approximations of both velocity and pressure. In addition, the fully discrete case in combination with higher order continuous Galerkin-Petrov (cGP) methods is studied. Error estimates of order k + 1 are proved, where k denotes the polynomial degree in time, assuming that the convective term is time-independent. Numerical results show that the predicted order is also achieved in the general case of time-dependent convective terms.
- ItemFinite element methods for the incompressible Stokes equations with variable viscosity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) John, Volker; Kaiser, Kristine; Novo, JuliaFinite element error estimates are derived for the incompressible Stokes equations with variable viscosity. The ratio of the supremum and the infimum of the viscosity appears in the error bounds. Numerical studies show that this ratio can be observed sometimes. However, often the numerical results show a weaker dependency on the viscosity.
- ItemA comparative study of a direct discretization and an operator-splitting solver for population balance systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Anker, Felix; Ganesan, Sashikumaar; John, Volker; Schmeyer, EllenA direct discretization approach and an operator-splitting scheme are applied for the numerical simulation of a population balance system which models the synthesis of urea with a uni-variate population. The problem is formulated in axisymmetric form and the setup is chosen such that a steady state is reached. Both solvers are assessed with respect to the accuracy of the results, where experimental data are used for comparison, and the efficiency of the simulations. Depending on the goal of simulations, to track the evolution of the process accurately or to reach the steady state fast, recommendations for the choice of the solver are given.
- ItemSome analytical results for an algebraic flux correction scheme for a steady convection-diffusion equation in 1D(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Barrenechea, Gabriel R.; John, Volker; Knobloch, PetrAlgebraic flux correction schemes are nonlinear discretizations of convection dominated problems. In this work, a scheme from this class is studied for a steady-state convection-diffusion equation in one dimension. It is proved that this scheme satisfies the discrete maximum principle. Also, as it is a nonlinear scheme, the solvability of the linear subproblems arising in a Picard iteration is studied, where positive and negative results are proved. Furthermore, the non-existence of solutions for the nonlinear scheme is proved by means of counterexamples. Therefore, a modification of the method, which ensures the existence of a solution, is proposed. A weak version of the discrete maximum principle is proved for this modified method.