Consider the numerical solution for Advection equation,

Where U > 0a) We use Taylor series and truncate the higher order derivatives.

Forward Difference in time, is given as,

Central Difference in space is given as,

Substituting in the Advection equation, we get,

where

b) Now, by Fourier series, error is given as,

Substituting in the Finite Difference equation, we get,

We know that, and . Substituting in above equation, we get

c) For stability, amplification factor G should always be less than or equal to 1.

But we know that . Hence, .

d) Applying Backward difference in space

Substituting in the Advection equation, we get,

where

e) Now, by Fourier series, error is given as,

Substituting in the Finite Difference equation, we get,

For Stability

This implies that

And

Hence this method is stable for