Week 1: Introduction to ODE's. Derivation of finite difference approximations. Initial value ODE. Accuracy, stability and convergence of numerical schemes.
Week 2: Multipoint methods for initial value ODE's. Stiff ODE. System of initial value ODE's .
Week 3: Boundary value ODE. Shooting and Equilibrium methods. Eigenvalue problems.
Week 4: Classification of PDE's. Characteristic variables and compatibility conditions. Initial and boundary conditions.
Week 5: The linear convection equation: explicit schemes, diffusion and dispersion errors, stability analysis.
Week 6: Methods for parabolic PDE's. The diffusion equation and convection-diffusion equation: explicit methods, stability analysis, boundary conditions.
Week 7: The convection-diffusion equation (continued): implicit methods, multidimensional problems.
Week 8: Methods for elliptic PDE's. Laplace and Poisson equations: direct solution, iterative solution, ADI method.
Week 9: Methods for elliptic PDE's. Iterative solutions: continued. The multigrid method.