Finite-volume Method - General Conservation Law

General Conservation Law

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

Here, represents a vector of states and represents the corresponding flux tensor. Again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, we take the volume integral over the total volume of the cell, 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 represents the total surface area of the cell and is a unit vector normal to the surface and pointing outward. So, finally, we are able to present the general result equivalent to (7), 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!

Read more about this topic:  Finite-volume Method

Famous quotes containing the words general, conservation and/or law:

    “The General Strike has taught the working class more in four days than years of talking could have done.”
    —A.J. (Arthur James)

    “The putting into force of laws which shall secure the conservation of our resources, as far as they may be within the jurisdiction of the Federal Government, including the more important work of saving and restoring our forests and the great improvement of waterways, are all proper government functions which must involve large expenditure if properly performed.”
    —William Howard Taft (1857–1930)

    “You are, or you are not the President of The National University Law School. If you are its President I wish to say to you that I have been passed through the curriculum of study of that school, and am entitled to, and demand my Diploma. If you are not its President then I ask you to take your name from its papers, and not hold out to the world to be what you are not.”
    —Belva Lockwood (1830–1917)