Sebastian Noelle | RWTH Aachen University (original) (raw)

Papers by Sebastian Noelle

cscamm.umd.edu

Abstract. A good numerical method for the Saint-Venant system of shallow water equations must be ... more Abstract. A good numerical method for the Saint-Venant system of shallow water equations must be well-balanced in the sense that the method should exactly preserve lake at rest steady states. There are many numerical methods capable of achieving this goal, but only in the ...

Over the last decade, the development of high resolution well-balanced schemes was a central topi... more Over the last decade, the development of high resolution well-balanced schemes was a central topic in the numerical analysis of hyperbolic systems. Indeed, many applications in continuum mechanics lead to systems of balance laws. For such flows, the source terms are often in near-perfect equilibrium with the convective forces. A numerical scheme which does not respect these equilibria at the discrete level may produce spurious oscillations, and hence convergence may slow down.

In this paper the authors review some recent work on high-order well-balanced schemes. A characte... more In this paper the authors review some recent work on high-order well-balanced schemes. A characteristic feature of hyperbolic systems of balance laws is the existence of non-trivial equilibrium solutions, where the effects of convective fluxes and source terms cancel each other. Well-balanced schemes satisfy a discrete analogue of this balance and are therefore able to maintain an equilibrium state. They

Siam Journal on Scientific Computing, 2000

. We present a new second-order, nonoscillatory, central dierence scheme ontwo-dimensional, stagg... more . We present a new second-order, nonoscillatory, central dierence scheme ontwo-dimensional, staggered, Cartesian grids for systems of conservation laws. The schemeuses a new, carefully designed integration rule for the ux computations and thereby takesmore propagation directions into account. This eectively reduces grid orientation eectsproduced for two-dimensional, radially symmetric gas ows.AMS subject classication (1991): 65M06, 35L65, 35L671. IntroductionNon-oscillatory,...

Hyperbolic Problems: Theory, Numerics, Applications, 2001

The use of staggered grids leads to Riemann-solver-free schemes for conservation laws. Here we de... more The use of staggered grids leads to Riemann-solver-free schemes for conservation laws. Here we develop a new adaptive staggered scheme. Given a locally refined cartesian grid based on a quadtree data-structure, we define the corresponding dual grid using a set of natural design-principles. The data on the dual grid can be stored and accessed efficiently via a hashtable. We present

We present a new second-order, nonoscillatory, central difference scheme on two-dimensional, stag... more We present a new second-order, nonoscillatory, central difference scheme on two-dimensional, staggered, Cartesian grids for systems of conservation laws. The scheme uses a new, carefully designed integration rule for the flux computations and thereby takes more propagation directions into account. This effectively reduces grid orientation effects produced for two-dimensional radially symmetric gas flows and improves the accuracy for smooth solutions.

SIAM Journal on Scientific Computing, 2003

The recently proposed high-order central difference schemes for conservation laws have a tendency... more The recently proposed high-order central difference schemes for conservation laws have a tendency of smearing linear discontinuities. In principle, Harten's artificial compression method (ACM) could be used to improve resolution. We analyze why this approach has not yet been used successfully and derive a more powerful version of the ACM based on a rigorous estimate of the total variation. We discuss the potential danger of overcompression and point out directions of future algorithmic development.

SIAM Journal on Scientific Computing, 2001

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation... more We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720-742].

Ocean Modelling, 2007

In this paper we compare a classical finite-difference and a high order finitevolume scheme for b... more In this paper we compare a classical finite-difference and a high order finitevolume scheme for barotropic ocean flows. We compare the schemes with respect to their accuracy, stability, and study various outflow and inflow boundary conditions. We apply the schemes to the problem of eddy formation in shelf slope jets along the Ormen Lange section of the Norwegian shelf. Our results strongly confirm the development of mesoscale eddies caused by instability of the flows.

Numerische Mathematik, 1995

We prove convergence of a class of higher order upwind finite volume schemes on unstructured grid... more We prove convergence of a class of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions. The result is applied to the discontinuous Galerkin method due to Cockburn, Hou and Shu.

Journal of Scientific Computing, 2011

This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-wa... more This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh

Journal of Fluid Mechanics, 2003

Shock waves, dead zones and particle-free regions form when a thin surface avalanche of granular ... more Shock waves, dead zones and particle-free regions form when a thin surface avalanche of granular material flows around an obstacle or over a change in the bed topography. Understanding and modelling these flows is of considerable practical interest for industrial processes, as well as for the design of defences to protect buildings, structures and people from snow avalanches, debris flows and rockfalls. These flow phenomena also yield useful constitutive information that can be used to improve existing avalanche models. In this paper a simple hydraulic theory, first suggested in the Russian literature, is generalized to model quasi-two-dimensional flows around obstacles. Exact and numerical solutions are then compared with laboratory experiments. These indicate that the theory is adequate to quantitatively describe the formation of normal shocks, oblique shocks, dead zones and granular vacua. Such features are generated by the flow around a pyramidal obstacle, which is typical of some of the defensive structures in use today.

Journal of Computational Physics, 2002

Shock formations are observed in granular avalanches when supercritical flow merges into a region... more Shock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shock-capturing numerical scheme for the one-dimensional Savage-Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a front-tracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the front-tracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat run-out region, where a shock is formed as the avalanche comes to a halt. c 2002 Elsevier Science

Journal of Computational Physics, 2007

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but a... more Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source term. In our earlier work [J. Comput. Phys. 208 (2005) 206-227; J. Sci. Comput., accepted], we designed a wellbalanced finite difference weighted essentially non-oscillatory (WENO) scheme, which at the same time maintains genuine high order accuracy for general solutions, to a class of hyperbolic systems with separable source terms including the shallow water equations, the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. In this paper, we generalize high order finite volume WENO schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property. Finite volume and discontinuous Galerkin finite element schemes are more flexible than finite difference schemes to treat complicated geometry and adaptivity. However, because of a different computational framework, the maintenance of the well-balanced property requires different technical approaches. After the description of our well-balanced high order finite volume WENO and RKDG schemes, we perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the nonoscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.

Journal of Computational Physics, 2006

Many geophysical flows are merely perturbations of some fundamental equilibrium state. If a numer... more Many geophysical flows are merely perturbations of some fundamental equilibrium state. If a numerical scheme shall capture such flows efficiently, it should be able to preserve the unperturbed equilibrium state at the discrete level. Here, we present a class of schemes of any desired order of accuracy which preserve the lake at rest perfectly. These schemes should have an impact for studying important classes of lake and ocean flows.

International Journal for Numerical Methods in Fluids, 2005

We present a new well-balanced finite volume method within the framework of the finite volume evo... more We present a new well-balanced finite volume method within the framework of the finite volume evolution Galerkin (FVEG) schemes. The methodology will be illustrated for the shallow water equations with source terms modelling the bottom topography and Coriolis forces. Results can be generalized to more complex systems of balance laws. The FVEG methods couple a finite volume formulation with approximate evolution operators. The latter are constructed using the bicharacteristics of multidimensional hyperbolic systems, such that all of the infinitely many directions of wave propagation are taken into account explicitly. We derive a wellbalanced approximation of the integral equations and prove that the FVEG scheme is well-balanced for the stationary steady states as well as for the steady jets in the rotational frame. Several numerical experiments for stationary and quasi-stationary states as well as for steady jets confirm the reliability of the well-balanced FVEG scheme.

International Journal for Numerical Methods in Biomedical Engineering, 2010

Page 1. ON ADAPTIVE TIMESTEPPING FOR WEAKLY INSTATIONARY SOLUTIONS OF HYPERBOLIC CONSERVATION LAW... more Page 1. ON ADAPTIVE TIMESTEPPING FOR WEAKLY INSTATIONARY SOLUTIONS OF HYPERBOLIC CONSERVATION LAWS VIA ADJOINT ERROR CONTROL CHRISTINA STEINER AND SEBASTIAN NOELLE Abstract. ... 1 Page 2. 2 Christina Steiner and Sebastian Noelle ...

ESAIM: Mathematical Modelling and Numerical Analysis, 2004

... Tim Kröger 1 , Sebastian Noelle 1 and Susanne Zimmermann 2 Abstract. ... We would like to emp... more ... Tim Kröger 1 , Sebastian Noelle 1 and Susanne Zimmermann 2 Abstract. ... We would like to emphasize that this is different from the fluctuation splitting approach of Deconinck, Roe and Struijs [7] where the divergence ∇x · F(U) is decomposed rather than the flux matrix F itself. ...

Applied Numerical Mathematics, 2006

Central schemes are frequently used for incompressible and compressible flow calculations. The pr... more Central schemes are frequently used for incompressible and compressible flow calculations. The present paper is the first in a forthcoming series where a new approach to a 2nd order accurate Finite Volume scheme operating on Cartesian grids is discussed. Here we start with an adaptively refined Cartesian primal grid in 3D and present a construction technique for the staggered dual grid based on L ∞ -Voronoi cells. The local refinement constellation on the primal grid leads to a finite number of uniquely defined local patterns on a primal cell. Assembling adjacent local patterns forms the dual grid. All local patterns can be analysed in advance. Later, running the numerical scheme on staggered grids, all necessary geometric information can instantly be retrieved from lookup-tables. The new scheme is compared to established ones in terms of algorithmic complexity and computational effort.