The finite volume method (FVM) is a numerical approach applicable to differential equations that represent various types of conservation laws. As a method it was originally developed to solve water related problems and has been extended to other disciplines later. FVM has the advantage that it can be used on arbitrary geometries and on structured or unstructured discretization of the computational domain.
In the FDM approach, detailed in the previous chapter, the differential equation to be solved is written at each discretization node of the computational domain by replacing the derivatives with a finite difference approximation. The FDM method, however is difficult to apply when the coefficients involved in the differential equation are discontinuous (LeVeque, 2002). In the FVM method, discontinuities of the coefficients are not a problem if the computational grid is chosen in such a way that these appear at the boundaries of the computational domain (Durran, 1998). Finite difference approximations are sometimes used in FVM approach to approximate fluxes at the boundary of the control volume, the discretization unit of the computational domain. Hence the FVM differs from the FDM in the way finite difference approximation are used (i.e., for the flux of a quantity, rather than for a derivative).
Depending on how the discretization of the computational domain is done, the FVM is also known as cell centred, vertex centred, box method, or balance method.
Finite volume methods have been extensively developed by different researchers for each type of partial differential equation (hyperbolic, parabolic and elliptic) and dependent of the discretization type used;
Lazarov and Makarov (1982) for Cartesian meshes, Mandel and McCormick (1989) for unstructured meshes; Manteuffel and White (1987) for cell-centred finite volume schemes; LeVeque (2002), Godlewski and Raviart (1996) for hyperbolic problems; and so on.
Finite volume methods can be characterised as follows:
• Problem domain is discretized into discrete elements called control volumes (CV);
• The differential equation to be solved is integrated into a balance equations for each control volume of the computational domain;
• Obtained integrals are approximated using numerical integration;
Chapter 5
Finite volume method
78 Computational Hydraulics
• The values of the unknown function and their derivatives are approximated using the values at the nodes of the control volume;
• The solution of the problem over the computational domain is obtained by assembling the obtained discrete equations over a control volume, into an algebraic system of equations.
An important feature of the FVM approach is the fact that the numerical flux is conserved from one discretization cell to another (the neighbouring cell), that is, there is local conservation of the numerical fluxes. This is very important especially because flux is an important element in fluid dynamics.
The FVM principle is detailed bellow. The method is applicable for the solution of conservation laws, and for simplicity of the demonstration of the method the one-dimensional general form of the conservation law (see chapter 2) is used:
∂t∂ u x t dV f u dA S u t dV
V A V
( , ) ( ) ( , )
∫
+∫
⋅ ⋅n −∫
= 0 (5.1)Equation (5.1) shows the transport of a variable (substance, water depth, etc.) under the influence of advection, that is, the rate of change of variable u(x, t) inside a defined volume V together with the flux of u(x, t) through boundary A is the same as the rate of S(u, t). Equation (5.1) is valid at each point x and each time t, of the computational domain, where the conservation of u(x, t) is written. S(u, x, t) is also known as source term of the equation and expresses a possible volumetric exchange between conserved quantities u(x, t), that is, the diffusion phenomena. The term n represents the unit vector perpendicular on surface A, pointing outwards.
Before showing the principle of the FVM method it is important to note that in the integral expression (5.1) the derivatives are zero order for the advection term ( f(u)) and first order for the diffusion term (S(x, t)), if it exists. The fact that the order of the derivatives is so low is important when solutions are sought for problems where phenomena change rapidly in space so that the spatial derivative does not exist, such as in the case of a hydraulic jump. Functions that are discontinuous do not have derivatives at the discontinuity position, hence the conservation equation expressed as a PDE, is not valid at the discontinuity point. Finite volume method presents the advantage that they overcome this issues, by considering the integral representation of the PDE and by having reduced order of derivatives, that can be easily treated.
The principle of the method is illustrated on a simple example of the one-dimensional linear advection equation, which is obtained from (5.1) if f(u) = au and S(u, t) = 0:
Consider the above equation to be solved over the one dimensional computational domain V that is discretized in control volumes Vi, defined between points xi−1/2 and xi+1/2 (see Figure 5.1b). Volume integration of equation (5.2) over the control volume (CV) yields:
∂∂ ∂ In Figure 5.1 points xi are equally spaced along the space domain x with a constant distance Δx = xi+1/2 − xi−1/2; i = 1, . . . , N − 1. Same convention of notation holds, as in Chapter 4, that u(xi+1) = u(xi + Δx) = ui+1. In Figure 5.1 control volumes are noted with Vi.
Finite volume method 79
... x x
1D Cell centered FVM 1D Vertex centered FVM
2D Vertex centered FVM 2D Cell centered FVM
xi-1 xi-1/2 xi
Figure 5.1 Finite volume examples of control volume in one- and two-dimensional space.
Equation (5.3) represents the integral formulation of the differential equation over control volume Vi which is written in all CVs of the computational domain to yield a system of equations, and gives the solution to the problem by determining the values of the unknown variable in all CVs.
Solution of the system formed by types of equation (5.3) requires:
• numerical approximation of the space integrals, which can be volume and/or surface integrals, depending on the problem dimension;
• interpolation, to determine the values of coefficients that describe the numerical approximations of the surface and volume integrals; and
• time integration by evaluating the time derivatives.
If ui is averaged and considered constant over CV, then the first term of the equation (5.3) can be integrated numerically as: The second term of equation (5.3) integrated analytically yields:
∂∂f Substitution of equation (5.4) and (5.5) into equation (5.3) gives its discrete form as:
ui⋅(xi+1 2/ + xi−1 2/ )+ f u( i+1 2/ )− f u( i−1 2/ ) = 0 (5.6) Equation (5.6) is a conservative scheme if the flux on the boundary of one cell is equal with the flux on the boundary of the neighboring cell.
80 Computational Hydraulics
The discretization of the space domain, as selected, is known as cell-centered method, where the unknown variables, u(x, t) are computed as average values over the CV and they are attached to the middle of the cell. A similar discretization is the cell-vertex method, in which the variables are averaged over a CV, but they are attached to the nodes of the discretization grid, that is, at the edge of the CV (see Figure 5.1a). Figure 5.1c, d, shows two types of discretization for a two dimensional space.