Repeat Problem 7.1 for the following schemes applied to the one-dimensional heat equation.

a) Simple explicit scheme \((7.28)\)

b) Simple implicit scheme \((7.32)\)

c) Crank-Nicolson scheme (7.34)

Problem 7.1

Compare the schemes introduced  for the linear convection equation.

a) Simple explicit schemes (7.6), (7.7), and (7.8)

b) Simple implicit scheme (7.19)

c) Leapfrog scheme (7.21)

d) Lax-Wendroff scheme \((7.24)\)

e) MacCormack scheme (7.26)

For every scheme, describe its stability properties (write the stability criterion, where appropriate) and order of approximation. Determine, where data are available, whether the numerical dissipation or numerical dispersion dominates the truncation error. Summarize the discussion stating, for every scheme, how appropriate it is in your opinion for solution of the linear convection equation. Does the scheme accurately reproduce the traveling wave solution shown in Figure 7.1? What are the main advantages and disadvantages of the scheme?