Finite volume method

Navier–Stokes differential equations used to simulate airflow around an obstruction.

Scope

Natural sciences Engineering
Astronomy Physics Chemistry Biology Geology
Applied mathematics
Continuum mechanics Chaos theory Dynamical systems
Social sciences
Economics Population dynamics

Classification

Types

By variable type
Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear
Dependent and independent variables Autonomous Complex Coupled / Decoupled Exact Homogeneous / Nonhomogeneous
Features
Order Operator Notation

Relation to processes

Difference (discrete analogue)

Solution

General topics

Lyapunov / Asymptotic / Exponential stability
Rate of convergence
Series / Integral solutions

Solution methods

People

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations [LeVeque, 2002; Toro, 1999]. Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence 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 are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many computational fluid dynamics packages.

1D example

Consider a simple 1D advection problem defined by the following partial differential equation

\quad (1)\qquad \qquad {\frac {\partial \rho }{\partial t}}+{\frac {\partial f}{\partial x}}=0,\quad t\geq 0.

Here, $\rho =\rho \left(x,t\right)\$ represents the state variable and $f=f\left(\rho \left(x,t\right)\right)\$ represents the flux or flow of $\rho \$ . Conventionally, positive $f\$ represents flow to the right while negative $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, $x\$ , into finite volumes or cells with cell centres indexed as $i\$ . For a particular cell, $i\$ , we can define the volume average value of ${\rho }_{i}\left(t\right)=\rho \left(x,t\right)\$ at time ${t=t_{1}}\$ and ${x\in \left[x_{{i-{\frac {1}{2}}}},x_{{i+{\frac {1}{2}}}}\right]}\$ , as

\quad (2)\qquad \qquad {\bar {\rho }}_{i}\left(t_{1}\right)={\frac {1}{x_{{i+{\frac {1}{2}}}}-x_{{i-{\frac {1}{2}}}}}}\int _{{x_{{i-{\frac {1}{2}}}}}}^{{x_{{i+{\frac {1}{2}}}}}}\rho \left(x,t_{1}\right)\,dx,

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

\quad (3)\qquad \qquad {\bar {\rho }}_{i}\left(t_{2}\right)={\frac {1}{x_{{i+{\frac {1}{2}}}}-x_{{i-{\frac {1}{2}}}}}}\int _{{x_{{i-{\frac {1}{2}}}}}}^{{x_{{i+{\frac {1}{2}}}}}}\rho \left(x,t_{2}\right)\,dx,

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

Integrating equation (1) in time, we have:

\quad (4)\qquad \qquad \rho \left(x,t_{2}\right)=\rho \left(x,t_{1}\right)-\int _{{t_{1}}}^{{t_{2}}}f_{x}\left(x,t\right)\,dt,

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

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

\quad (5)\qquad \qquad {\bar {\rho }}_{{i}}\left(t_{{2}}\right)={\frac {1}{\Delta x_{i}}}\int _{{x_{{i-{\frac {1}{2}}}}}}^{{x_{{i+{\frac {1}{2}}}}}}\left\{\rho \left(x,t_{{1}}\right)-\int _{{t_{{1}}}}^{{t_{2}}}f_{{x}}\left(x,t\right)dt\right\}dx.

We assume that $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 $f_{x}\triangleq \nabla \cdot f$ , we can apply the divergence theorem, i.e. $\oint _{{v}}\nabla \cdot fdv=\oint _{{S}}f\,dS$ , and substitute for the volume integral of the divergence with the values of $f(x) \$ evaluated at the cell surface (edges $x_{{i-{\frac {1}{2}}}}\$ and $x_{{i+{\frac {1}{2}}}}\$ ) of the finite volume as follows:

\quad (6)\qquad \qquad {\bar {\rho }}_{i}\left(t_{2}\right)={\bar {\rho }}_{i}\left(t_{1}\right)-{\frac {1}{\Delta x_{{i}}}}\left(\int _{{t_{1}}}^{{t_{2}}}f_{{i+{\frac {1}{2}}}}dt-\int _{{t_{1}}}^{{t_{2}}}f_{{i-{\frac {1}{2}}}}dt\right).

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

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

\quad (7)\qquad \qquad {\frac {d{\bar {\rho }}_{i}}{dt}}+{\frac {1}{\Delta x_{i}}}\left[f_{{i+{\frac {1}{2}}}}-f_{{i-{\frac {1}{2}}}}\right]=0,

where values for the edge fluxes, $f_{{i\pm {\frac {1}{2}}}}$ , can be reconstructed by interpolation or extrapolation 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.

General conservation law

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

\quad (8)\qquad \qquad {{\partial {{\mathbf u}}} \over {\partial t}}+\nabla \cdot {{\mathbf f}}\left({{\mathbf u}}\right)={{\mathbf 0}}.

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

\quad (9)\qquad \qquad \int _{{v_{{i}}}}{{\partial {{\mathbf u}}} \over {\partial t}}\,dv+\int _{{v_{{i}}}}\nabla \cdot {{\mathbf f}}\left({{\mathbf u}}\right)\,dv={{\mathbf 0}}.

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

\quad (10)\qquad \qquad v_{{i}}{{d{{\mathbf {{\bar u}}}}_{{i}}} \over {dt}}+\oint _{{S_{{i}}}}{{\mathbf f}}\left({{\mathbf u}}\right)\cdot {{\mathbf n}}\ dS={{\mathbf 0}},

where $S_{{i}}\$ represents the total surface area of the cell and ${{\mathbf n}}$ 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.

\quad (11)\qquad \qquad {{d{{\mathbf {{\bar u}}}}_{{i}}} \over {dt}}+{{1} \over {v_{{i}}}}\oint _{{S_{{i}}}}{{\mathbf f}}\left({{\mathbf u}}\right)\cdot {{\mathbf n}}\ dS={{\mathbf 0}}.

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 another cell's gain!

References

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.
LeVeque, Randall (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.
Toro, E. F. (1999), Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag.

External links

The finite volume method 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

Numerical partial differential equations by method

Finite difference

Parabolic	Forward-time central-space (FTCS) Crank–Nicolson

Hyperbolic	Lax–Friedrichs Lax–Wendroff MacCormack Upwind Method of characteristics

Others	Alternating direction-implicit (ADI) Finite-difference time-domain (FDTD)

Finite volume

Finite element

Meshless/Meshfree

Smoothed-particle hydrodynamics (SPH)
Moving Particle Semi-implicit Method (MPS)
Material point method (MPM)

Domain decomposition

Others

This article is issued from Wikipedia - version of the 8/27/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Finite volume method

1D example

General conservation law

See also

References

Further reading

External links