Since , the right inequality is always true. For the left inequality, we have
which holds irrespective of the value of under the condition
Problem 2
Question (a)
The stencil is shown below.
Limits on and are
Question (b)
FDE for the Dufort-Frankel scheme is
Extract and use Taylor expansion ( subscript is omitted), we have
Use and drop higher order terms, we have
This is of order . Due to the presence of term, the method is not consistent when , where when . Therefore the method is only conditionally consistent.
Question (c)
The method is explicit. can be directly calculated from , and without solving a linear system or inverse function.
Question (d)
Use notation, the FDE can be written as
Extract , we have
Substitute with the generic solution , we have
Assume , and divide both sides by , we have
which yields the following 2nd order equation
The roots of this equation are
For the method to be stable, we need . For real roots where , we have
For complex roots where , we have
and
Note that guarantees that , and the method is unconditionally stable.
Question (e)
From , we have
This indicates that though deacys exponentially, it does so slowly when is large. Therefore, the method is stable but not L-stable.
Problem 3
Question (a)
From equation (3.22)
we can replace terms with Taylor expansion at , and obtain
Cancel duplicated terms, we have
where has an order of . Note that the fifth spatial derivative terms in also cancel out, so the remain in is instead of .
