Méthode des volumes finis

Résumé

En analyse numérique, la méthode des volumes finis est utilisée pour résoudre numériquement des équations aux dérivées partielles, comme la méthode des différences finies et celle des éléments finis. Contrairement à la méthode des différences finies, qui met en jeu des approximations des dérivées, les méthodes des volumes finis et des éléments finis exploitent des approximations d'intégrales. Toutefois, la méthode des volumes finis exploite directement la forme dite forte de l'équation à résoudre, alors que la méthode des éléments finis se fonde sur une formulation variationnelle de l'équation (on parle aussi de formulation faible). L'équation aux dérivées partielles est résolue de manière approchée à l’aide d’un maillage constitué de volumes finis, qui sont de petits volumes disjoints en 3D (des surfaces en 2D, des segments en 1D) dont la réunion constitue le domaine d'étude. Les volumes finis peuvent être construits autour de points d'un maillage initial, mais ce n’est pas une néc

Source officielle

https://fr.wikipedia.org/wiki/M%C3%A9thode_des_volumes_finis

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Cours associés (14)

This course provides practical experience in the numerical simulation of fluid flows. Numerical methods are presented in the framework of the finite volume method. A simple solver is developed with Matlab, and a commercial software is used for more complex problems.

Ce cours présente une introduction aux méthodes d'approximation utilisées pour la simulation numérique en mécanique des fluides. Les concepts fondamentaux sont présentés dans le cadre de la méthode des différences finies puis étendus à celles des éléments finis et spectraux.

Introduction to the development, analysis, and application of computational methods for solving conservation laws with an emphasis on finite volume, limiter based schemes, high-order essentially non-oscillatory schemes, and discontinuous Galerkin methods.

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Séances de cours associées (33)

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High-order essentially nonoscillatory methods based on radial basis functions

Fabian Mönkeberg

Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely employed to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose high-order essentially nonoscillatory methods based on radial basis functions (RBFs) to solve hyperbolic conservation laws. We derive a smoothness indicator that guarantees the satisfaction of the sign property of the resulting interpolant on general one-dimensional grids. Based on this algorithm we introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Hence, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. Next, we propose a high-order ENO method based on radial basis functions to solve hyperbolic conservation laws on general two-dimensional unstructured grids. The radial basis function reconstruction offers a flexible framework to deal with ill-conditioned cell arrangements. We define a smoothness indicator based the one-dimensional version and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting to circumvent the stagnation of the error and keep the condition number of the reconstruction bounded. To reduce the computational complexity, we develop the RBF-CWENO method. This method exhibits high-order convergence and robustness when solving challenging problems and is considerably faster. However, the resolution close to shocks and turbulent structures is lower than for the RBF-ENO method. Finally, we present a hybrid high-resolution RBF-ENO method which is based on the RBF-ENO method for unstructured patches and the standard WENO method on structured ones. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods that stabilizes the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness on the scramjet inflow problem and a conical aerospike nozzle jet simulation.

EPFL2020

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This paper presents a mesh adaptation procedure linked to a finite volume solver, the goal of which is to increase the precision of the numerical simulation of a wing tip vortex flow. The adaptation scheme is applied to hexahedron meshes and hybrid meshes made up of tetrahedrons and prisms. To evaluate the ability of each type of element to capture the physics of a tip vortex, a specific test case is studied and results obtained numerically from this test case are compared with experimental results. The error estimator of the adaptation scheme is derived from a solution scalar variable. It is shown that the element anisotropy as well as the adaptation algorithms used have an impact on the precision of the solution. Adaptation of hexahedrons allows a better capture of the tip vortex far from the vortex root, even though the adaptation of those hexahedrons barely changes the number of nodes used to achieve a specified precision, contrary to the adaptation of hybrid meshes. Copyright (C) 2009 John Wiley & Sons, Ltd.

2011

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A 1D Model For A Closed Rigid Filament In Stokes Flow

Philippe Caussignac

We adapt an existing asymptotic method to set up a one-dimensional model for the fall of a closed filament in an in finite fluid in the Stokes regime. Starting from the single-layer integral representation of the fluid velocity around the filament, we get, for a very slender filament, a Fredholm integral equation on the filament centerline. From this equation, we can compute the drag and force acting on the filament and consequently the resistance matrix. The integral equation is discretized with a collocation method. The study of a scalar model problem yields existence and uniqueness results together with an error estimate for the discretization scheme. Then, we compare the resistance matrix of thin ideal knots obtained from the discretization of the present model to a boundary element method; numerical convergence results and a good agreement of both methods validate our model.

2009

Chargement

High-order essentially nonoscillatory methods based on radial basis functions

Fabian Mönkeberg

EPFL2020