In the text above we showed that the numerical approximations generated by the explicit scheme...

In the text above we showed that the numerical
approximations generated by the explicit scheme (7.91) satisfy the bound

The purpose of the present exercise is to prove a somewhat
stronger result. We will assume throughout this exercise that the
discretization parameters have been chosen such that

(a) Show that
the approximations generated by the scheme (7.91) satisfy

(b) Prove that

provided that the discretization
parameters satisfy (8.97). For notational purposes we introduce the symbols uN`
max and uN` min for the maximum and minimum values, respectively, of the
discrete approximations at time step

(c) Apply the
monotonicity property derived in (b) to prove that

(d) Show that

(e) Prove in a
similar manner that

which is a discrete analog to the
maximum principle (8.41)–(8.43) valid for the continuous problem (8.1)–(8.3).

8.43)

problem (8.1)–(8.3).

(f) Above
we mentioned that we would derive a stronger bound for the numerical
approximations of the solution of the diffusion equation than inequality
(8.96). Explain why (8.98) is a stronger result than (8.96), i.e., use (8.98)
to derive (8.96).