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:

    “We all have known
    Good critics, who have stamped out poet’s hopes;
    Good statesmen, who pulled ruin on the state;
    Good patriots, who, for a theory, risked a cause;
    Good kings, who disembowelled for a tax;
    Good Popes, who brought all good to jeopardy;
    Good Christians, who sat still in easy-chairs;
    And damned the general world for standing up.—
    Now, may the good God pardon all good men!”
    —Elizabeth Barrett Browning (1806–1861)

    “A country grows in history not only because of the heroism of its troops on the field of battle, it grows also when it turns to justice and to right for the conservation of its interests.”
    —Aristide Briand (1862–1932)

    “It grieves me to think how far more profound and reverent a respect the law would have for literature if a body could only get drunk on it.”
    —Mark Twain [Samuel Langhorne Clemens] (1835–1910)