FiniteVolumeMethods，有限体积法经典书目RobertEymard,ThierryGallou¨et，andRapha`eleHerbin资源-CSDN文库

Finite

Volume

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Finite Volume Methods

Robert Eymard

, Thierry Gallou¨et

and Rapha`ele Herbin

October 2006. 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

Ecole Nationale des Ponts et Chauss´ees, Marne-la-Vall´ee, et Universit´e de Paris XIII

Ecole Normale Sup´erieure de Lyon

Universit´e de Provence, Marseille

Contents

1 Introduction 4

1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The ﬁnite volume principles for general conservation laws . . . . . . . . . . . . . . . . . . 6

1.2.1 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 Space discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Comparison with other disc retization techniques . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 General guideline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 A one-dimensional elliptic problem 12

2.1 A ﬁnite volume method for the Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1 Formulation of a ﬁnite volume scheme . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Comparison with a ﬁnite diﬀerence scheme . . . . . . . . . . . . . . . . . . . . . . 14

2.1.3 Comparison with a mixed ﬁnite element method . . . . . . . . . . . . . . . . . . . 15

2.2 Converge nce theorems and error estimates for the Dirichlet problem . . . . . . . . . . . . 16

2.2.1 A ﬁnite volume error estimate in a simple case . . . . . . . . . . . . . . . . . . . . 16

2.2.2 An error estimate using ﬁnite diﬀerence techniques . . . . . . . . . . . . . . . . . . 19

2.3 General 1D elliptic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Formulation of the ﬁnite volume scheme . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.2 Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 The case of a point source term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 A semilinear elliptic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Problem and Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.2 Compactness results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.3 Converge nce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Elliptic problems in two or three dimensions 32

3.1 Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.1 Structured meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3

3.1.2 General meshes and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.1.3 Existence and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.4 Converge nce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1.5 C

error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.6 H

error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Neumann boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2.1 Meshes and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2.2 Discrete Poincar´e inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.2.3 Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.4 Converge nce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 General elliptic o perators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.1 Discontinuous matrix diﬀusion coeﬃcients . . . . . . . . . . . . . . . . . . . . . . . 77

Version de d´ecembre 2003 2

3.3.2 Other boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4 Dual meshes and unknowns located at vertices . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4.1 The piecewise linear ﬁnite element method viewed as a ﬁnite volume method . . . 84

3.4.2 Classical ﬁnite volumes on a dual mes h . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4.3 “Finite Volume Finite Element” methods . . . . . . . . . . . . . . . . . . . . . . . 88

3.4.4 Generalization to the three dimensional case . . . . . . . . . . . . . . . . . . . . . 90

3.5 Mesh reﬁnement and s ingularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5.1 Singular source terms and ﬁnite volumes . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5.2 Mesh reﬁnement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.6 Compactness results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 Parabolic equations 95

4.1 Meshes and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Error estimate for the linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.3 Converge nce in the nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3.1 Solutions to the continuous problem . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.3.2 Deﬁnition o f the ﬁnite volume approximate solutions . . . . . . . . . . . . . . . . . 103

4.3.3 Estimates on the approximate solution . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.3.4 Converge nce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.3.5 Weak convergence and nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.3.6 A uniquenes s res ult for nonlinear diﬀusion equations . . . . . . . . . . . . . . . . . 115

5 Hyperbolic equations in the one dimensional case 119

5.1 The continuous problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2 Numerical schemes in the linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2.1 The centered ﬁnite diﬀerence scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.2 The upstream ﬁnite diﬀerence scheme . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.2.3 The upwind ﬁnite volume scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 The nonlinear case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.1 Meshes and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.2 L

∞

-stability for monotone ﬂux schemes . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3.3 Discrete entropy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.3.4 Converge nce of the upstream scheme in the general case . . . . . . . . . . . . . . . 134

5.3.5 Converge nce proof using BV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.4 Higher order schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3

5.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.5.1 A general convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.5.2 A very simple example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6

5.5.3 A simplifed model for two phase ﬂows in pipelines . . . . . . . . . . . . . . . . . . 147

6 Multidimensional nonlinear hyperbol ic equations 150

6.1 The continuous problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.2 Meshes and schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.2.1 Explicit schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.2.2 Implicit schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5

6.2.3 Passing to the limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.3 Stability results for the explicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.3.1 L

∞

stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.3.2 A “weak BV ” es timate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.4 Existence of the solution and stability results for the implicit scheme . . . . . . . . . . . . 161

6.4.1 Existence, uniqueness and L

∞

stability . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4.2 “Weak space BV ” inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Version de d´ecembre 2003 3

6.4.3 “Time BV ” estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.5 Entropy inequalities for the approximate solution . . . . . . . . . . . . . . . . . . . . . . . 169

6.5.1 Discrete entropy inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.5.2 Continuous e ntropy estimates for the approximate solution . . . . . . . . . . . . . 171

6.6 Converge nce of the scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.6.1 Converge nce towards an entropy pro c e ss solution . . . . . . . . . . . . . . . . . . 179

6.6.2 Uniqueness of the entropy process solution . . . . . . . . . . . . . . . . . . . . . . 18 0

6.6.3 Converge nce towards the entropy weak solution . . . . . . . . . . . . . . . . . . . . 184

6.7 Error estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.7.1 Statement of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.7.2 Preliminary lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.7.3 Proof of the error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.7.4 Remarks and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.8 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

6.9 Nonlinear weak-⋆ convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.10 A stabilized ﬁnite element method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

6.11 Moving meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7 Systems 204

7.1 Hyperbolic systems of equatio ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

7.1.1 Classical schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

7.1.2 Rough s chemes for complex hyperbolic systems . . . . . . . . . . . . . . . . . . . . 207

7.1.3 Partial implicitation of explicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . 210

7.1.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11

7.1.5 Staggered grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7.2 Incompressible Navier-Stokes E quations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

7.2.1 The continuous equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

7.2.2 Structured stagger e d gr ids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6

7.2.3 A ﬁnite volume scheme on unstructured staggered grids . . . . . . . . . . . . . . . 216

7.3 Flows in porous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.3.1 Two phase ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

7.3.2 Compositional multiphase ﬂow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

7.3.3 A simpliﬁed cas e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

7.3.4 The scheme for the simpliﬁed case . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

7.3.5 Estimates on the approximate solution . . . . . . . . . . . . . . . . . . . . . . . . . 226

7.3.6 Theorem of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

7.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

7.4.1 A two phas e ﬂow in a pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

7.4.2 Two phase ﬂow in a p orous medium . . . . . . . . . . . . . . . . . . . . . . . . . . 236

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 s imilar to those of the ﬁnite element

method, see Oden [118]: it may be used on arbitrary geometries, using structured or unstructured

meshes, and it leads to robust schemes. An additional feature is the local conserva tivity of the numerical

ﬂuxes, tha t is the numerical ﬂux is conserved from o ne discretization cell to its neighbour. This last

feature makes the ﬁnite volume method quite attractive when modelling problems for which the ﬂux is of

impo rtance, such as in ﬂuid mechanics, semi-conductor device simulation, heat and mass transfer. . . The

ﬁnite volume method is locally conservative because it is based on a “ balance” approach: a local balance

is written on each discr e tization cell which is often called “control volume”; by the divergence formula,

an integral formulation of the ﬂuxes over the boundary of the control volume is then obtained. The ﬂuxes

on the boundary are discretized with respect to the discrete unknowns.

Let us introduce the method more precisely on simple ex amples, and then give a description of the

discretization of general conservation laws.

1.1 Examples

Two basic examples can be used to introduce the ﬁnite volume method. They will be develop e d in details

in the following chapters.

Example 1.1 (Transpo rt equation) Consider ﬁrst the linea r transport equation



(x, t) + div(vu)(x, t) = 0, x ∈ IR

, t ∈ IR

u(x, 0) = u

(x), x ∈ IR

(1.1)

where u

denotes the time derivative of u, v ∈ C

(IR

, IR

), and u

∈ L

∞

(IR

). Let T be a mesh of

consisting of polygonal bounded convex subsets of IR

and let K ∈ T be a “control volume”, that

is an element of the mesh T . Integrating the ﬁrst equation of (1.1) over K yields the following “balance

equation” over K:

(x, t)dx +

∂K

v(x, t) · n

(x)u(x, t)dγ(x) = 0, ∀t ∈ IR

, (1.2)

where n

denotes the normal vector to ∂K, outward to K. Let k ∈ IR

∗

be a cons tant time discretization

step and let t

= nk, for n ∈ IN. Writing equation (1.2) at time t

, n ∈ IN and discretizing the time

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