Thefinite volume method (FVM) is a method for representing and evaluatingpartial differential equations in the form of algebraic equations.^[1]In the finite volume method, volume integrals in a partial differential equation that contain adivergence term are converted tosurface integrals, using thedivergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods areconservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in manycomputational fluid dynamics packages."Finite volume" refers to the small volume surrounding each node point on a mesh.^[2]

Finite volume methods can be compared and contrasted with thefinite difference methods, which approximate derivatives using nodal values, orfinite element methods, which create local approximations of a solution using local data, and construct a global approximation by stitching them together. In contrast a finite volume method evaluates exact expressions for theaverage value of the solution over some volume, and uses this data to construct approximations of the solution within cells.^[3][4]

Consider a simple 1Dadvection problem:

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\frac{\mathrm{\partial }f}{\mathrm{\partial }x}=0,t\ge 0.}$

1

Here,${\displaystyle \rho =\rho \left(x,t\right)}$ represents the state variable and${\displaystyle f=f\left(\rho \left(x,t\right)\right)}$ represents theflux or flow of${\displaystyle \rho }$. Conventionally, positive${\displaystyle f}$ represents flow to the right while negative${\displaystyle f}$ represents flow to the left. If we assume that equation (1) represents a flowing medium of constant area, we can sub-divide the spatial domain,${\displaystyle x}$, intofinite volumes orcells with cell centers indexed as${\displaystyle i}$. For a particular cell,${\displaystyle i}$, we can define thevolume average value of${\displaystyle {\rho }_i\left(t\right)=\rho \left(x,t\right)}$ at time${\displaystyle t=t_1}$ and${\displaystyle x\in \left[x_{i-1/2},x_{i+1/2}\right]}$, as

${\displaystyle {\overline{\rho }}_i\left(t_1\right)=\frac{1}{x_{i+1/2}-x_{i-1/2}}{\int }_{x_{i-1/2}}^{x_{i+1/2}}\rho \left(x,t_1\right)dx,}$

2

and at time${\displaystyle t=t_2}$ as,

${\displaystyle {\overline{\rho }}_i\left(t_2\right)=\frac{1}{x_{i+1/2}-x_{i-1/2}}{\int }_{x_{i-1/2}}^{x_{i+1/2}}\rho \left(x,t_2\right)dx,}$

3

where${\displaystyle x_{i-1/2}}$ and${\displaystyle x_{i+1/2}}$ represent locations of the upstream and downstream faces or edges respectively of the${\displaystyle i^{\text{th}}}$ cell.

Integrating equation (1) in time, we have:

${\displaystyle \rho \left(x,t_2\right)=\rho \left(x,t_1\right)-{\int }_{t_1}^{t_2}{f}_x\left(x,t\right)dt,}$

4

where${\displaystyle f_x=\frac{\mathrm{\partial }f}{\mathrm{\partial }x}}$.

To obtain the volume average of${\displaystyle \rho \left(x,t\right)}$ at time${\displaystyle t=t_2}$, we integrate${\displaystyle \rho \left(x,t_2\right)}$ over the cell volume,${\displaystyle \left[x_{i-1/2},x_{i+1/2}\right]}$ and divide the result by${\displaystyle \mathrm{\Delta }{x}_i=x_{i+1/2}-x_{i-1/2}}$, i.e.

${\displaystyle {\overline{\rho }}_i\left(t_2\right)=\frac{1}{\mathrm{\Delta }{x}_i}{\int }_{x_{i-1/2}}^{x_{i+1/2}}\left\{\rho \left(x,t_1\right)-{\int }_{t_1}^{t_2}{f}_x\left(x,t\right)dt\right\}dx.}$

5

We assume that${\displaystyle f}$ is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the unit area of the cell. Now, since in one dimension${\displaystyle f_x\triangleq \mathrm{\nabla }\cdot f}$, we can apply thedivergence theorem, i.e.${\displaystyle {\oint }_v\mathrm{\nabla }\cdot fdv={\oint }_{S}fdS}$, and substitute for the volume integral of thedivergence with the values of${\displaystyle f(x)}$ evaluated at the cell surface (edges${\displaystyle x_{i-1/2}}$ and${\displaystyle x_{i+1/2}}$) of the finite volume as follows:

${\displaystyle {\overline{\rho }}_i\left(t_2\right)={\overline{\rho }}_i\left(t_1\right)-\frac{1}{\mathrm{\Delta }{x}_i}\left({\int }_{t_1}^{t_2}{f}_{i+1/2}dt-{\int }_{t_1}^{t_2}{f}_{i-1/2}dt\right).}$

6

where${\displaystyle f_{i\pm 1/2}=f\left(x_{i\pm 1/2},t\right)}$.

We can therefore derive asemi-discrete numerical scheme for the above problem with cell centers indexed as${\displaystyle i}$, and with cell edge fluxes indexed as${\displaystyle i\pm 1/2}$, by differentiating (6) with respect to time to obtain:

${\displaystyle \frac{d{\overline{\rho }}_i}{dt}+\frac{1}{\mathrm{\Delta }{x}_i}\left[f_{i+1/2}-f_{i-1/2}\right]=0,}$

7

where values for the edge fluxes,${\displaystyle f_{i\pm 1/2}}$, can be reconstructed byinterpolation orextrapolation of the cell averages. Equation (7) isexact for the volume averages; i.e., no approximations have been made during its derivation.

This method can also be applied to a2D situation by considering the north and south faces along with the east and west faces around a node.

We can also consider the generalconservation law problem, represented by the followingPDE,

${\displaystyle \frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathbf{f}\left(\mathbf{u}\right)=\mathbf{0}.}$

8

Here,${\displaystyle \mathbf{u}}$ represents a vector of states and${\displaystyle \mathbf{f}}$ represents the correspondingflux tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell,${\displaystyle i}$, we take the volume integral over the total volume of the cell,${\displaystyle v_i}$, which gives,

${\displaystyle {\int }_{v_i}\frac{\mathrm{\partial }\mathbf{u}}{\mathrm{\partial }t}dv+{\int }_{v_i}\mathrm{\nabla }\cdot \mathbf{f}\left(\mathbf{u}\right)dv=\mathbf{0}.}$

9

On integrating the first term to get thevolume average and applying thedivergence theorem to the second, this yields

${\displaystyle v_i\frac{d{\overline{\mathbf{u}}}_i}{dt}+{\oint }_{S_i}\mathbf{f}\left(\mathbf{u}\right)\cdot \mathbf{n}dS=\mathbf{0},}$

10

where${\displaystyle S_i}$ represents the total surface area of the cell and${\displaystyle \mathbf{n}}$ is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (8), i.e.

${\displaystyle \frac{d{\overline{\mathbf{u}}}_i}{dt}+\frac{1}{v_i}{\oint }_{S_i}\mathbf{f}\left(\mathbf{u}\right)\cdot \mathbf{n}dS=\mathbf{0}.}$

11

Again, values for the edge fluxes can be reconstructed by interpolation or extrapolation of the cell averages. The actual numerical scheme will depend upon problem geometry and mesh construction.MUSCL reconstruction is often used inhigh resolution schemes where shocks or discontinuities are present in the solution.

Finite volume schemes are conservative as cell averages change through the edge fluxes. In other words,one cell's loss is always another cell's gain!

• Finite element method
• Flux limiter
• Godunov's scheme
• Godunov's theorem
• High-resolution scheme
• KIVA (software)
• MIT General Circulation Model
• MUSCL scheme
• Sergei K. Godunov
• Total variation diminishing
• Finite volume method for unsteady flow

• Eymard, R. Gallouët, T. R.,Herbin, R. (2000)The finite volume method Handbook of Numerical Analysis, Vol. VII, 2000, p. 713–1020. Editors: P.G. Ciarlet and J.L. Lions.
• Hirsch, C. (1990),Numerical Computation of Internal and External Flows, Volume 2: Computational Methods for Inviscid and Viscous Flows, Wiley.
• Laney, Culbert B. (1998),Computational Gas Dynamics, Cambridge University Press.
• LeVeque, Randall (1990),Numerical Methods for Conservation Laws, ETH Lectures in Mathematics Series, Birkhauser-Verlag.
• LeVeque, Randall (2002),Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
• Patankar, Suhas V. (1980),Numerical Heat Transfer and Fluid Flow, Hemisphere.
• Tannehill, John C., et al., (1997),Computational Fluid mechanics and Heat Transfer, 2nd Ed., Taylor and Francis.
• Toro, E. F. (1999),Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.
• Wesseling, Pieter (2001),Principles of Computational Fluid Dynamics, Springer-Verlag.

• Finite volume methods by R. Eymard, T Gallouët andR. Herbin, update of the article published in Handbook of Numerical Analysis, 2000
• Rübenkönig, Oliver."The Finite Volume Method (FVM) – An introduction". Archived fromthe original on 2009-10-02., available under theGFDL.
• FiPy: A Finite Volume PDE Solver Using Python from NIST.
• CLAWPACK: a software package designed to compute numerical solutions to hyperbolic partial differential equations using a wave propagation approach

• Numerical differential equations
• Computational fluid dynamics
• Numerical analysis

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