Finite volume method

Consider a simple 1D advection problem:

Here, ${\displaystyle \rho =\rho \left(x,t\right)}$  represents the state variable and ${\displaystyle f=f\left(\rho \left(x,t\right)\right)}$  represents the flux 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, ${\displaystyle x}$ , into finite volumesorcells with cell centers indexed as ${\displaystyle i}$ . For a particular cell, ${\displaystyle i}$ , we can define the volume 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

and at time ${\displaystyle t=t_{2}}$  as,

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:

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

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 ${\displaystyle \Delta x_{i}=x_{i+1/2}-x_{i-1/2}}$ , i.e.

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 ${\displaystyle f_{x}\triangleq \nabla \cdot f}$ , we can apply the divergence theorem, i.e. ${\displaystyle \oint _{v}\nabla \cdot fdv=\oint _{S}f\,dS}$ , and substitute for the volume integral of the divergence 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:

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

We can therefore derive a semi-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:

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

This method can also be applied to a 2D situation by considering the north and south faces along with the east and west faces around a node.

We can also consider the general conservation law problem, represented by the following PDE,

Here, ${\displaystyle \mathbf {u} }$  represents a vector of states and ${\displaystyle \mathbf {f} }$  represents the corresponding flux 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,

On integrating the first term to get the volume average and applying the divergence theorem to the second, this yields

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.

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 in high 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.
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• 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 and R. Herbin, update of the article published in Handbook of Numerical Analysis, 2000
• Rübenkönig, Oliver. "The Finite Volume Method (FVM) – An introduction". Archived from the original on 2009-10-02., available under the GFDL.
• 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