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3 Material Balance Finite Difference Equations in One Dimension     continued...

Combining the approximation for the change in storage, Eq. 12, with the respective approximations for net flux into the control interval, Eq. 13, Eq. 14, and Eq. 15, yields:

Explicit Method:

Implicit Method:

Midpoint Method:

The final step in arriving at a finite difference equation that can be used in a computational setting is to relate the fluxes at and to the field variable u.

First, consider the case that is a constant. Using centered finite difference formulas to approximate the fluxes at the control points and yields

and

with similar formulas for the fluxes at time .