Difference Equations and Conservation

Assignment :

Study and understand the Equations in the PDF file
Do Exercise 4.

This lecture is an extension of the last one, as I look at implications of finite volume and finite difference approximations. We will begin with a review of the equations, expand discussion of the Courant stability limit, and discuss positivity of cell mass and conservation of system mass.

Look at the Difference Equation Summary (Postscript), be sure you learn the difference between Fully Implicit, Fully Explicit, Crank-Nicholson, and Semi-Implicit. Remember to think in terms of a staggered mesh with thermodynamic variables defined at volume centers and velocities at volume edges. Pay careful attention to the definition of the finite volume divergence operator. We will use it shortly in discussions of Conservative difference equations. Memorize the definitions of averages for Donor-Cell (Upwind) and Central difference methods. Finally take note of the pattern of use of new and old time values in the Semi-Implicit form of the Euler flow equations.

Conservation of mass says that the integral of density over the entire volume of the system should only change by the net amount fluxed into or out of the system. This can be obtained by integrating the differential equation for conservation of mass over the full system. For our types of systems results are best when a similar result can be obtained by summing the finite volume equation over all volumes in the system.

Stability of Equations

The Courant stability limit applies to an Explicit scheme because it can't propagate any information, including sound wave more than one cell in one step. For example, when you solve the explicit finite volume mass equation for new time density, the right hand side of the equation only contains information about density in the adjacent cell. Assume a straight pipe ( cell volume equal to cell length times cell area), and assume a positive flow velocity. Think about a situation where the time step is greater than the cell length divided by the sum of sound speed and magnitude of the flow velocity. In the real system this is enough time for information (pressure wave) to travel from cell j-2 to cell j. However the equations for change in properties in cell j, contain no information about cell j-2. You expect strange things to happen, and they do (instabilities as in your home work).

The Semi-Implicit scheme propagates sound waves over the entire system in each step. So we don't worry about sound speed in stability considerations. However a Material Courant stability limit applies, because the method can't propagate mass, energy or momentum flux information more than one cell in one step. Time step must be less than cell length divided by the magnitude of velocity.

Positive Density (and Energy)

In addition to basic stability, one would generally like a numerical solution to preserve basic physical ranges for properties, such as positive cell mass and internal energy. As long as the material Courant stability limit is obeyed, a rearrangement of a Donor-Cell difference equation shows that density will stay positive (assume positive velocity)

If velocity changes sign, across a cell (say negative at face j-1/2 and positive at j+1/2), then problems could arise with negative densities.

A central difference method also has potential problems with negative densities if a sharp enough density gradient exists.

Two-Phase Considerations

Energy Conservation

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