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Finite Volume Methods Robert Eymard, Thierry Gallouët, Raphaèle Herbin

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Finite Volume Methods Robert Eymard, Thierry Gallouët, Raphaèle Herbin Finite Volume Methods Robert Eymard, Thierry Gallouët, Raphaèle Herbin To cite this version: Robert Eymard, Thierry Gallouët, Raphaèle Herbin. Finite Volume Methods. J. L. Lions; Philippe Ciarlet. Solution of Equation in n (Part 3), Techniques of Scientific Computing (Part 3), 7, Elsevier, pp.713-1020, 2000, Handbook of Numerical Analysis, 9780444503503. 10.1016/S1570-8659(00)07005- 8. hal-02100732v2 HAL Id: hal-02100732 https://hal.archives-ouvertes.fr/hal-02100732v2 Submitted on 12 Aug 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Finite Volume Methods Robert Eymard1, Thierry Gallou¨et2 and Rapha`ele Herbin3 January 2019. This manuscript is an update of the preprint n0 97-19 du LATP, UMR 6632, Marseille, September 1997 which appeared in Handbook of Numerical Analysis, P.G. Ciarlet, J.L. Lions eds, vol 7, pp 713-1020 1Ecole Nationale des Ponts et Chauss´ees, Marne-la-Vall´ee, et Universit´ede Paris XIII 2Ecole Normale Sup´erieure de Lyon 3Universit´ede Provence, Marseille Contents 1 Introduction 4 1 Examples ........................................... .. 4 2 The finite volume principles for general conservation laws . ........... 6 2.1 Time discretization . 7 2.2 Spacediscretization ................................. .. 8 3 Comparison with other discretization techniques . .......... 8 4 Generalguideline..................................... .... 9 2 A one-dimensional elliptic problem 12 5 A finite volume method for the Dirichlet problem . ....... 12 5.1 Formulation of a finite volume scheme . ... 12 5.2 Comparison with a finite difference scheme . .... 14 5.3 Comparison with a mixed finite element method . ... 15 6 Convergence and error analysis for the Dirichlet problem . ............ 16 6.1 Error estimate with C2 regularity............................ 16 6.2 An error estimate using a finite difference technique . ....... 19 7 General 1D elliptic equations . ..... 21 7.1 Formulation of the finite volume scheme . .... 21 7.2 Errorestimate ..................................... 23 7.3 Thecaseofapointsourceterm. .. .. .. .. .. .. .. .. .. .. .... 26 8 A semilinear elliptic problem . .. 27 8.1 ProblemandScheme.................................. 27 8.2 Compactnessresults ................................ ... 28 8.3 Convergence ...................................... 30 3 Elliptic problems in two or three dimensions 32 9 Dirichlet boundary conditions . ...... 32 9.1 Structuredmeshes ................................. ... 33 9.2 Generalmeshesandschemes .......................... .... 37 9.3 Existenceandestimates .............................. ... 42 9.4 Convergence ...................................... 44 9.5 C2 errorestimate..................................... 52 9.6 H2 errorestimate .................................... 55 10 Neumannboundaryconditions . ....... 63 10.1 Meshesandschemes ................................ ... 63 10.2 Discrete Poincar´einequality . ..... 65 10.3 Errorestimate .................................... .. 69 10.4 Convergence ..................................... .. 72 11 Generalellipticoperators ............................. ....... 78 11.1 Discontinuous matrix diffusion coefficients . ...... 78 1 Version de janvier 2019 2 11.2 Otherboundaryconditions . .. .. .. .. .. .. .. .. .. .. .. .... 82 12 Dualmeshesandunknownslocatedatvertices . ........... 84 12.1 The piecewise linear finite element method viewed as a finite volume method . 84 12.2 Classical finite volumes on a dual mesh . .... 86 12.3 “Finite Volume Finite Element” methods . ... 89 12.4 Generalization to the three dimensional case . ....... 90 13 Meshrefinementandsingularities. ........ 91 13.1 Singular source terms and finite volumes . ...... 91 13.2 Meshrefinement ................................... .. 93 14 Compactnessresults ................................ ....... 94 4 Parabolic equations 97 15 Introduction...................................... ...... 97 16 Meshesandschemes ................................. ...... 98 17 Errorestimateforthelinearcase . ......... 100 18 Convergenceinthenonlinearcase . ........ 104 18.1 Solutions to the continuous problem . ..... 104 18.2 Definition of the finite volume approximate solutions . ....... 105 18.3 Estimates on the approximate solution . ...... 106 18.4 Convergence ..................................... 113 18.5 Weak convergence and nonlinearities . ...... 116 18.6 A uniqueness result for nonlinear diffusion equations . ........ 118 5 Hyperbolic equations in the one dimensional case 122 19 Thecontinuousproblem ............................... ...... 122 20 Numericalschemesinthelinearcase . ........ 125 20.1 Thecenteredfinitedifferencescheme . ....... 126 20.2 Theupstreamfinitedifferencescheme . ...... 126 20.3 Theupwindfinitevolumescheme. 128 21 Thenonlinearcase ................................... ..... 133 21.1 Meshesandschemes ................................ 133 21.2 L∞-stabilityformonotonefluxschemes . 136 21.3 Discrete entropy inequalities . .. 136 21.4 Convergence of the upstream scheme in the general case . ............ 137 21.5 Convergence proof using BV .............................. 141 22 Higherorderschemes................................ ....... 146 23 Boundaryconditions ................................. ...... 147 23.1 Ageneralconvergenceresult . ...... 147 23.2 Averysimpleexample................................. 149 23.3 A simplifed model for two phase flows in pipelines . .. 150 6 Multidimensional nonlinear hyperbolic equations 153 24 Thecontinuousproblem ............................... ...... 153 25 Meshesandschemes ................................. ...... 156 25.1 Explicitschemes .................................... 157 25.2 Implicitschemes .................................... 158 25.3 Passing to the limit . 158 26 Stability results for the explicit scheme . ......... 160 26.1 L∞ stability........................................ 160 26.2 A “weak BV ”estimate ................................. 161 27 Existence of the solution and stability results for the implicit scheme............ 164 27.1 Existence, uniqueness and L∞ stability ........................ 164 Version de janvier 2019 3 27.2 “Weak space BV ”inequality .............................. 167 27.3 “Time BV ”estimate................................... 168 28 Entropy inequalities for the approximate solution . ........... 172 28.1 Discrete entropy inequalities . .. 172 28.2 Continuous entropy estimates for the approximate solution . ........... 174 29 Convergenceofthescheme .. .. .. .. .. .. .. .. .. .. .. ........ 181 29.1 Convergencetowards an entropy process solution . .......... 182 29.2 Uniquenessoftheentropyprocesssolution . ........ 183 29.3 Convergencetowards the entropy weak solution . .......... 188 30 Errorestimate ..................................... ..... 189 30.1 Statementoftheresults ............................ ..... 189 30.2 Preliminary lemmata . 191 30.3 Proofoftheerrorestimates . ...... 197 30.4 Remarksandopenproblems ........................... 199 31 Boundaryconditions ................................. ...... 199 32 Nonlinear weak-⋆ convergence.................................. 201 33 A stabilized finite element method . ...... 204 34 Movingmeshes ...................................... .. 205 7 Systems 208 35 Hyperbolicsystemsofequations . ........ 208 35.1 Classicalschemes.................................. 209 35.2 Roughschemesforcomplexhyperbolicsystems . ........ 211 35.3 Partial implicitation of explicit scheme . .. 214 35.4 Boundaryconditions................................ 215 35.5 Staggeredgrids................................... 218 36 IncompressibleNavier-StokesEquations . ........... 219 36.1 Thecontinuousequation............................. .. 219 36.2 Structuredstaggeredgrids. ....... 220 36.3 A finite volume scheme on unstructured staggered grids . .......... 220 37 Flowsinporousmedia.................................. .. 223 37.1 Twophaseflow..................................... 223 37.2 Compositional multiphase flow . 224 37.3 Asimplifiedcase ..................................... 226 37.4 Theschemeforthesimplifiedcase . .. 227 37.5 Estimates on the approximate solution . ...... 230 37.6 Theoremofconvergence .. .. .. .. .. .. .. .. .. .. .. .. ..... 233 38 Boundaryconditions ................................. ...... 237 38.1 A two phase flow in a pipeline . 238 38.2 Twophaseflowinaporousmedium . 240 Bibliography Chapter 1 Introduction The finite volume method is a discretization method which is well suited for the numerical simulation of various types (elliptic, parabolic or hyperbolic, for instance) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer or petroleum engineer- ing. Some of the important features of the finite volume method are similar to those of the finite element method, see Oden [121]: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. An additional feature is the local conservativity of the numerical fluxes, that is the numerical flux is conserved from one discretization cell to its neighbour.
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