In a project below, we will derive further methods for solving initial value problems of the form where f is a given function and u0 is the known initial state. In this exercise, we will derive three...

In a project below, we will derive further methods for solving initial value problems of the form

where f is a given function and u0 is the known initial state. In this exercise, we will derive three schemes based on formulas for numerical integration. Suppose we want to solve (2.57) from
 where, as usual,

and N >0 is an integer.

(a)    Show that

(b)   Use the trapezoidal rule to motivate the following numerical scheme

(c)    Use the midpoint method to motivate the scheme

We will derive this scheme below, using another approach. The scheme is often called the Crank–Nicolson scheme, but it really studies a somewhat different problem.

(d)   Use Simpson’s scheme to motivate

(e)   Implement the schemes given by (2.59)–(2.61). Use problem (2.38) to check the accuracy of the schemes.