Finite Elements and characteristics for some parabolic-hyperbolic problems

Bercovier, Michel; Pironneau, Olivier; Sastri, V. M. K.

doi:10.1016/0307-904x(83)90118-x

Cited by 74 publications

(28 citation statements)

References 9 publications

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“…n, = 0 are given as the boundary conditions. Strp 7: Repeat steps 3-6 using the working equation (20) instead of equation (24) and obtain the values of water depth at all nodal points. The only boundary conditions prescribed in this step are the water depths, as a function of time, at the down-stream side of the flow domain.…”

Section: Solution Algorithmmentioning

confidence: 99%

A New Two-Dimensional Finite Element Model for the Shallow Water Equations Using a Lagrangian Framework Constructed Along Fluid Particle Trajectories

Petera

Nassehi

1996

Int. J. Numer. Meth. Engng.

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SUMMARYIn this paper we describe a new finite element model for the tidal hydrodynamics in estuaries. The mathematical model is based on the solution of the two-dimensional shallow water equations in a Lagrangian framework which is defined along the trajectories of fluid particles. This method gives a flexible and robust numerical scheme for moving boundary flows encountered in tidal water systems. In order to validate the developed model we have. at first instance, compared our numerical results with analytical solutions obtained for domains with simple geometries. Further tests are then conducted to demonstrate the model's ability to cope with conditions such as hydraulic shock, abrupt changes in the flow domain geometry and gradual changes of water surface breadth. The change in the water surface breadth corresponds to the drying and wetting of the plains along the banks of a typical tidal riverlestuary reach. The drying and wetting of flood plains result in the existence of very shallow depth of water at some sections of the flow domain during a tidal cycle. The flow equations under these conditions are strongly convection dominated. Previously published tidal models rely on either, some form of upwinding or the use o f extremely fine meshes to give stable results for the convection dominated very shallow depth computations in estuaries. We show that our model can yield stable and accurate results for very shallow depths in the tidal flow domains without using any kind of artifical damping or excessive mesh refinement. Computational costs of simulating hydrodynamical conditions in a natural water course, even using a depth averaged two-dimensional approach, can be very high. The ability of our scheme to cope with convection dominated conditions has enabled us to economize the computational efforts by using coarse meshes in our finite element calculations. After the validation stage. the developed model is applied to simulate the tidal conditions in a real estuary. The comparison of the model results with the field observations shows a close agreement between these sets of data KEY WORDS Tay cstunry shallow water equations. finite element method; Lagrangian framewoik; fluid parlicle trajectories:

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Section: Solution Algorithmmentioning

confidence: 99%

A New Two-Dimensional Finite Element Model for the Shallow Water Equations Using a Lagrangian Framework Constructed Along Fluid Particle Trajectories

Petera

Nassehi

1996

Int. J. Numer. Meth. Engng.

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“…Then the Lagrangian total derivative is approximated thanks to some divided difference operator. These methods have largely proven their efficiency (see for example [1,2,6,15,16] Let T > 0. We consider a (not necessarily uniform) subdivision of (0, T ): 0 = T 0 < T 1 < ... < T Q−1 < T Q = T .…”

Section: Introductionmentioning

confidence: 99%

An Adaptive Multi-level method for Convection Diffusion Problems

Marion

Mollard

2000

ESAIM: M2AN

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Abstract. In this article we introduce an adaptive multi-level method in space and time for convection diffusion problems. The scheme is based on a multi-level spatial splitting and the use of different timesteps. The temporal discretization relies on the characteristics method. We derive an a posteriori error estimate and design a corresponding adaptive algorithm. The efficiency of the multi-level method is illustrated by numerical experiments, in particular for a convection-dominated problem.Mathematics Subject Classification. 65M15, 65M25, 65M70, 76R99.

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“…In this problem, A is an artificial velocity and V • A is not necessary nuil so we will have to take it into consideration and compute it. Next, we shall follow the notation and ideas presented in [4].…”

Section: Numerical Analysis Of the Radially Symmetrical Case In 2dmentioning

confidence: 99%

“…(10), weopt for using a scheme which combines the method of characteristics and the reformulation of the convection term in (10) in terms of the total derivative with the finite element one (see [4,8] and references therein). We shall assume that the variables satisfy some regularity requirements that allow us to develop the numerical scheme.…”

Section: Numerical Analysis Of the Radially Symmetrical Case In 2dmentioning

confidence: 99%

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Numerical resolution of a reinforced random walk model arising in haptotaxis

Muñoz

Tello

2015

Applied Mathematics and Computation

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In this paper we study the numerical resolution of a reinforced random walk model arising in haptotaxis and the stabilization of solutions. The model consists of a system of two differential equations, one parabolic equation with a second order non-linear term (haptotaxis term) coupled to an ODE in a bounded two dimensional domain. We assume radial symmetry of the solutions. The scheme of resolution is based on the application of the characteristics method together with a finite element one. We present some numerical simulations which ¡Ilústrate some features of the numerical stabilization of solutions.

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Finite Elements and characteristics for some parabolic-hyperbolic problems

Cited by 74 publications

References 9 publications

A New Two-Dimensional Finite Element Model for the Shallow Water Equations Using a Lagrangian Framework Constructed Along Fluid Particle Trajectories

A New Two-Dimensional Finite Element Model for the Shallow Water Equations Using a Lagrangian Framework Constructed Along Fluid Particle Trajectories

An Adaptive Multi-level method for Convection Diffusion Problems

Numerical resolution of a reinforced random walk model arising in haptotaxis

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