Two-dimensional modelling of the river Rhine

Schulz, Martin; Steinebach, Gerd

doi:10.1016/s0377-0427(01)00532-5

Cited by 13 publications

(10 citation statements)

References 7 publications

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“…and the remainder term R being defined according to (33) and (21), with L given by (28), respectively.…”

Section: Remark 32mentioning

confidence: 99%

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Adaptive modeling for free-surface flows

Perotto¹

2006

ESAIM: M2AN

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Abstract. This work represents a first step towards the simulation of the motion of water in a complex hydrodynamic configuration, such as a channel network or a river delta, by means of a suitable "combination" of different mathematical models. In this framework a wide spectrum of space and time scales is involved due to the presence of physical phenomena of different nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solve throughout the whole domain the most complex model (with solution u fine ) to accurately describe all the physical features of the problem at hand. In our approach instead, for a user-defined output functional F, we aim to approximate, within a prescribed tolerance τ , the value F(u fine ) by means of the quantity F(u adapted ), u adapted being the so-called adapted solution solving the simpler models on most of the computational domain while confining the complex ones only on a restricted region. Moving from the simplified setting where only two hydrodynamic models, fine and coarse, are considered, we provide an efficient tool able to automatically select the regions of the domain where the coarse model rather than the fine one are to be solved, while guaranteeing |F(u fine ) − F(u adapted )| below the tolerance τ . This goal is achieved via a suitable a posteriori modeling error analysis developed in the framework of a goal-oriented theory. We extend the dual-based approach provided in [Braack and Ern, Multiscale Model Sim. 1 (2003) 221-238], for steady equations to the case of a generic time-dependent problem. Then this analysis is specialized to the case we are interested in, i.e. the free-surface flows simulation, by emphasizing the crucial issue of the time discretization for both the primal and the dual problems. Finally, in the last part of the paper a widespread numerical validation is carried out.Mathematics Subject Classification. 65J15, 65M15, 65M60.

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“…and the remainder term R being defined according to (33) and (21), with L given by (28), respectively.…”

Section: Remark 32mentioning

confidence: 99%

“…However, this approach requires a huge computational effort. In order to reduce such a computational cost, a hierarchy of simplified hydrodynamic models has been proposed in the literature (see, e.g., [33,36,37]). …”

Section: Introductionmentioning

confidence: 99%

Adaptive modeling for free-surface flows

Perotto¹

2006

ESAIM: M2AN

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“…This choice turns out not to be so restrictive in realistic situations. Indeed, even if the cross-section is irregular, a sophisticated channel schematization can be employed by resorting to rectangular sections (see Schulz and Steinebach [2002]). In such a case the 1D model reduces to the system…”

Section: The 1d Modelmentioning

confidence: 99%

A Multiphysics Strategy for Free Surface Flows

Miglio¹,

Perotto²,

Saleri³

2005

Lecture Notes in Computational Science and Engineering

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Summary. This work is the first step towards a multiphysics strategy for freesurface flows simulation. In particular, we present a strategy to couple one and two-dimensional hydrostatic free surface flow models. We aim to reduce the computational cost required by a full 2D model. After introducing the two models along with suitable a priori error estimates, we discuss the choice of convenient matching conditions stemming from the results obtained in Formaggia et al. [2001]. The numerical results in the last section confirm the soundness of our analysis.

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“…The past few decades have witnessed intensive research effort in interface problems [2, 6, 7, 17, 21, 26, 39, 36, 41, 42, 44, 47, 51, 54, 63, 64, 65, 66, 58, 72, 10]. These references are just the tip of a iceberg of whole literature.…”

Section: Introductionmentioning

confidence: 99%

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

Xia

Wei

2014

Computers & Mathematics with Applications

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We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the elements, so that the differentiation operations of the original PDEs can be discretized as if there was no interface. The extra coefficients of polynomial basis functions, which furnish the overlapping elements and solve the fictitious solutions, are determined by interface jump conditions. Consequently, the interface jump conditions are rigorously enforced on the interface. The present method utilizes Cartesian meshes to avoid the mesh generation in conventional finite element methods (FEMs). We implement the proposed MIB Galerkin method with three different elements, namely, rectangular prism element, five-tetrahedron element and six-tetrahedron element, which tile the Cartesian mesh without introducing any new node. The accuracy, stability and robustness of the proposed 3D MIB Galerkin are extensively validated over three types of elliptic interface problems. In the first type, interfaces are analytically defined by level set functions. These interfaces are relatively simple but admit geometric singularities. In the second type, interfaces are defined by protein surfaces, which are truly arbitrarily complex. The last type of interfaces originates from multiprotein complexes, such as molecular motors. Near second order accuracy has been confirmed for all of these problems. To our knowledge, it is the first time for an FEM to show a near second order convergence in solving the Poisson equation with realistic protein surfaces. Additionally, the present work offers the first known near second order accurate method for C1 continuous or H2 continuous solutions associated with a Lipschitz continuous interface in a 3D setting.

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Two-dimensional modelling of the river Rhine

Cited by 13 publications

References 7 publications

Adaptive modeling for free-surface flows

Adaptive modeling for free-surface flows

A Multiphysics Strategy for Free Surface Flows

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems

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