Explicit (Forward in Time) Finite Difference Method [for ]:
or, after minor algebraic rearrangement,
This FDE is referred to as explicit since Eq. 21 provides an explicit formula for calculating the solution to the difference equation at time , knowing the values at time . A finite difference scheme can be described to be stable if rounding errors do not grow faster than the solution as successive time steps are advanced. It can be shown that Eq. 21 is a stable computational method only if .
If we evaluate the numerical approximations to the flux at time , we obtain the following.
Implicit (Backward in Time) Finite Difference Method [for ]:
which can be written
Note that Eq. 22 only implicitly defines the solution at time , since a system of algebraic equations is required to be satisfied. The solution of this system of equations is discussed in the next section. The implicit method Eq. 22 can be shown to be stable for all values of r. The development of the Crank--Nicolson FDE from Eq. 18 is left to the exercises.