Matthias Möller

Algebraic Flux Correction I. Scalar Conservation Laws

An algebraic approach to the design of multidimensional high-resolution schemes is introduced and elucidated in the finite element context. A centered space discretization of unstable convective terms is rendered local extremum diminishing by a conservative elimination of negative off-diagonal coefficients from the discrete transport operator. This modification leads to an upwind-biased low-order scheme which is nonoscillatory but overly diffusive. In order to reduce the incurred error, a limited amount of compensating antidiffusion is added in regions where the solution is sufficiently smooth. Two closely related flux correction strategies are presented. The first one is based on a multidimensional generalization of total variation diminishing (TVD) schemes, whereas the second one represents an extension of the FEM-FCT paradigm to implicit time-stepping. Nonlinear algebraic systems are solved by an iterative defect correction scheme preconditioned by the low-order evolution operator which enjoys the M-matrix property. The dffusive and antidiffusive terms are represented as a sum of antisymmetric internodal fluxes which are constructed edge-by-edge and inserted into the global defect vector. The new methodology is applied to scalar transport equations discretized in space by the Galerkin method. Its performance is illustrated by numerical examples for 2D benchmark problems.

Algebraic Flux Correction II. Compressible Euler Equations

Algebraic flux correction schemes of TVD and FCT type are extended to systems of hyperbolic conservation laws. The group finite element formulation is employed for the treatment of the compressible Euler equations. An efficient algorithm is proposed for the edge-by-edge matrix assembly. A generalization of Roe’s approximate Riemann solver is derived by rendering all off-diagonal matrix blocks positive semi-definite. Another usable low-order method is constructed by adding scalar artificial viscosity proportional to the spectral radius of the cumulative Roe matrix. The limiting of antidiffusive fluxes is performed using a transformation to the characteristic variables or a suitable synchronization of correction factors for the conservative ones. The outer defect correction loop is equipped with a block-diagonal preconditioner so as to decouple the discretized Euler equations and solve them in a segregated fashion. As an alternative, a strongly coupled solution strategy (global BiCGSTAB method with a block-Gaus-Seidel preconditioner) is introduced for applications which call for the use of large time steps. Various algorithmic aspects including the implementation of characteristic boundary conditions are addressed. Simulation results are presented for inviscid flows in a wide range of Mach numbers.

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L^2 projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L^2 projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.

Isogeometric Analysis of the Navier-Stokes equations with Taylor-Hood B-spline elements

This paper presents our numerical results of the application of Isogeometric Analysis (IGA) to the velocity-pressure formulation of the steady state as well as to the unsteady incompressible Navier-Stokes equations. For the approximation of the velocity and pressure fields, LBB compatible B-spline spaces are used which can be regarded as smooth generalizations of Taylor-Hood pairs of finite element spaces. The single-step ?-scheme is used for the discretization in time. The lid-driven cavity flow, in addition to its regularized version and flow around cylinder, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme.

Three-dimensional semi-idealized model for tidal motion in tidal estuaries

In this paper, a three-dimensional semi-idealized model for tidal motion in a tidal estuary of arbitrary shape and bathymetry is presented. This model aims at bridging the gap between idealized and complex models. The vertical profiles of the velocities are obtained analytically in terms of the first-order and the second-order partial derivatives of surface elevation, which itself follows from an elliptic partial differential equation. The surface elevation is computed numerically using the finite element method and its partial derivatives are obtained using various methods. The newly developed semi-idealized model allows for a systematic investigation of the influence of geometry and bathymetry on the tidal motion which was not possible in previously developed idealized models. The new model also retains the flexibility and computational efficiency of previous idealized models, essential for sensitivity analysis. As a first step, the accuracy of the semi-idealized model is investigated. To this end, an extensive comparison is made between the model results of the semi-idealized model and two other idealized models: a width-averaged model and a three-dimensional idealized model. Finally, the semi-idealized model is used to understand the influence of local geometrical effects on the tidal motion in the Ems estuary. The model shows that local convergence and meandering effects can have a significant influence on the tidal motion. Finally, the model is applied to the Ems estuary. The model results agree well with observations and results from a complex numerical model.

Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems

The development of adaptive numerical schemes for steady transport equations is addressed. A goal-oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation. The error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual problems. The Galerkin orthogonality error is taken into account and found to be important whenever flux or slope limiters are activated to enforce monotonicity constraints. The localization of global errors is performed using a natural decomposition of the involved weights into nodal contributions. A nodal generation function is employed in a hierarchical mesh adaptation procedure which makes each refinement step readily reversible. The developed simulation tools are applied to a linear convection problem in two space dimensions.

Implicit Flux-Corrected Transport Algorithm for Finite Element Simulation of the Compressible Euler Equations

Even today, the accurate treatment of convection-dominated transport problems remains a challenging task in numerical simulation of both compressible and incompressible flows. The discrepancy arises between high accuracy and good resolution of singularities on the one hand and preventing the growth and birth of nonphysical oscillations on the other hand. In 1959 it was proven [6], that linear methods are restricted to be at most first order if they are to preserve monotonicity. Thus, the use of nonlinear methods is indispensable to overcome smearing by numerical diffusion without sacrificing important properties of the exact solution such as positivity and monotonicity. The advent of the promising methodology of flux-corrected transport (FCT can be traced back to the pioneering work of Boris and Book [3]. Even though their original FCT algorithm named SHASTA was a rather specialized one-dimensional finite difference scheme, the cornerstone for a variety of high-resolution schemes was laid. Strictly speaking, the authors recommended using a high-order discretization in regions of smooth solutions and switching to a low-order method in the vicinity of steep gradients. This idea of adaptive toggling between methods of high and low order was dramatically improved by Zalesak [3] who proposed a multi-dimensional generalization applicable to arbitrary combinations of high- and low-order discretizations but still remaining in the realm of finite differences. This barrier was first crossed by Parrott and Christie [21] who settled the idea of flux-correction in the framework of finite elements. Finally, FEM-FCT reached maturity by the considerable contributions of Lohner and his coworkers [16], [17]. Beside the classical formulation of Zalesak’s limiter in terms of element contributions, an alternative approach is available limiting the fluxes edge-by-edge [26], [27].

Temporal oscillations in the simulation of foam enhanced oil recovery

Many enhanced oil recovery (EOR) processes can be described using partial differential equations with parameters that are strongly non-linear functions of one or more of the state variables. Typically these non-linearities result in solution components changing several orders of magnitude over small spatial or temporal distances. The numerical simulation of such processes with the aid of finite volume or finite element techniques poses challenges. In particular, temporally oscillating state variable values are observed for realistic grid sizes when conventional discretization schemes are used. These oscillations, which do not represent a physical process but are discretization artifacts, hamper the use of the forward simulation model for optimization purposes. To analyze these problems, we study the dynamics of a simple foam model describing the interaction of water, gas and surfactants in a porous medium. It contains sharp gradients due to the formation of foam. The simplicity of the model allows us to gain a better understanding of the underlying processes and difficulties of the problem. The foam equations are discretized by a first-order finite volume method. Instead of using a finite volume method with a standard interpolation procedure, we opt for an integral average, which smooths out the discontinuity caused by foam generation. We introduce this method by applying it to the heat equation with discontinuous thermal conductivity. A similar technique is then applied to the foam model, reducing the oscillations drastically, but not removing them.

Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192, 2012; J Comput Appl Math 236:2317–2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915–4946, 2004; Int J Numer Methods Fluids 42:265–295, 2003; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525–558, 2002; J Comput Phys 198:131–158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming