The Finite Volume Method

3.2.1 Preliminaries

There are different equivalent ways to develop the Finite Volume Method. Our approach is to rely mainly on Gauss (or Divergence) theorem and the derived integration by parts (Green’s Lemma) in order to always focus on the conservation principle that is the foundation of the models of interest. We recall here the most important mathematical results that will be of use in our developments, reminding to standard analysis books for the details of the proofs.

Theorem 3.1 (Gauss or Divergence). Given a subset Ω ∈ R^d having piecewise smooth boundary Γ = ∂Ω, and a continuously differentiable (C¹(Ω)) vector field F (x)∈ Ω, we have:

Z

Ω

div F dx = Z

Γ

F · ν ds, (3.4)

whereν is the outward unit normal to Γ, dx denotes the volume (surface) measure on Ω and ds the surface measure onΓ and v· w = hv, wi denotes the scalar product between vectors v, w of R^d.

Theorem 3.2 (First Green Identity – Green’s Lemma). Let v, w ∈ R^dbe piecewise continuous vector fields and letΩ and Γ as in the previous theorem, then:

Z

wherediv∇ = ∆ is Laplace differential operator of second derivatives.

Remark 3.2. In (3.4) the left-hand-side contains the sum of the partial derivatives of the vector field F , and for this reason the hypothesis of the theorem contains the requirement that F be continuously differentiable. The right-hand-side, on the other hand, does not contain any derivative, and thus in principle the integral of the fluxes over the subset Ω could be defined without the requirement that F beC¹. However, the normal to the surface Γ must be well-defined and this is the reason for the requirement that Γ be piecewise smooth. In fact, if the boundary Γ is formed by the union of m smooth surfaces that intersect at boundaries that formC¹curves, i.e.:

Γ = The same argumentation can be made, with the appropriate changes, for Green’s Lemma 3.5.

The Finite Volume (FV) scheme can be derived by the following operations:

1. partition the domain Ω into M polygonal “finite volumes” or cells T_i, i = 1, . . . , M (see next paragraph for a complete definition of cells);

2. integrate equation (3.1) in time and space over the domain Ω and the time interval [t^k, t^k+1]:

Z t^k+1

3. use the linearity property of the integration to write:

Z t^k+1

impose that each term of the sum is zero:

Z t^k+1 t^k

Z

Ti

(u_t+ div f (u) dx) dt = 0 i = 1, . . . , M ;

4. apply the divergence theorem and use the fact that the cells are of polygonal shape:

Z t^k+1

5. exchange the first space integral with the time integral (the functions are assumed to be contin-uous and the domain of integration is assumed to be constant):

Z

6. integrate the first addendum in time:

Z 7. define cell average and the edge flux operators as:

A^T(u(t)) = 1

we remark that until now we have made no numerical approximations;

9. start the numerical approximation very naturally by approximating the cell average u_h,i ≈ A^Ti(u) = A^h(u) and use u_h as unknown of our numerical scheme together with a simple quadrature rule (e.g. left rectangles) to evaluate the remaining time integral to obtain:

u^j+1_h,i = u^j_h,i− ∆t

Remark 3.3. The first few points of the derivation could have been replaced by a derivation more aware of the definition of a weak formulation given in the previous sections. To this aim we proceed as follows:

1. partition the domain Ω into M polygonal “finite volumes” or cells T_i, i = 1, . . . , M (see next paragraph for a complete definition of cells); define a piecewise smooth test function φ(x) given by the characteristic function of the i-th cell:

φ_i(x, t) = χ_x(x)χ_t(t), where χ_x(x) =

(1 if x∈ Ti

0 otherwise; χ_t(t) =

(1 if t∈ [t^k, t^k+1] 0 otherwise;

2. multiply equation (3.1) and integrate over the domain Ω and the interval [t^k, t^k+1]:

Z t^k+1

this equation must be satisfied for all functions φ_i(x, t):

Z t^k+1

At this point we are left with the task of defining the numerical flux G^j_h,j for each mesh edge. Before going into the development of how to evaluate the numerical flux we need to setup some notation. We would like to stress here that this general Finite Volume setting is what is known as “cell-based”. We could derive a “node-based” version more similar to the approach used in the Finite Element Method without adding any complication. Within the same framework we could tackle more complicated PDEs, as for example parabolic or elliptic equations, obtained for example by adding to (3.1) a dif-fusion term proportional to the second derivatives of u(x, t). For a general discussion on these topics we refer the reader to the specialized Finite Volume literature [5, e.g.] for a complete mathematical theory and to [6] for application to Computational Fluid Dynamics.

3.2.2 Notations

Geometrically we define the meshT^h(Ω) as a finite collection of non-overlapping and non-empty two-dimensional “control volumes” or “cells” generally formed by simplices (e.g, subintervals, triangles, tetrahedra in one-, two-, and three-dimensions, respectively) and denoted with the letter “T” indexed by a Latin subscript, e.g. i, j, k. For example, Ti is the i-th control volume (cell) of the mesh T^h ={Tⁱ}, i = 1, . . . , M, with M being the total number of cells. We assume that, for every possible

x j

FIGURE 3.1: One-dimensional Finite Volume mesh.

We identify with the symbol σ a mesh face, i.e., the intersection between two neighboring cells, and index it with the indices of the two cells:

σ_ij = T_i∩ T^j,

and with the symbol v the vertices forming the cells. In a two-dimensional example, two distinct cells are either neighbors, in which case their intersections is the common boundary “edge”, or they are far apart, in which case their intersection is empty. Cells, faces, and vertices are identified by global numbers and all are counted only once. Appendix C reports typical efficient data structures that can be used to completely describe the mesh and corresponding quantities in a computer program. For more details consult [11].

Different meshes are parameterized by h, called the “mesh parameter” and defined as the maximum face measure, and by the “mesh diameter” ρ, i.e., the maximum diameter of the circles inscribed in each mesh cell. Letting h_Tand ρ_Tbeing the maximum face measure and inscribed circle diameter for the generic cell T we have then:

h = max

T∈T_hh_T ρ = max

T∈Th

ρ_T.

We require the triangulation to be “regular”, i.e., there exists a constant β > 0 independent from h and ρ and from the triangulationTh such that:

ρT

h_T ≥ β for all T ∈ T^h.

The value of β is a measure of the minimum angle between two consecutive cell edges. The as-sumption that the triangulation is regular implies that cell angles do not become too small in the limiting process as h 7→ 0, so that the problem of numerical interpolation of nodal or edge values is well-posed.