Part I. Numerical Methods for Elliptic Problems

Finite-Difference Approximations

Steady-States and Boundary Value Problems

- a simple finite-difference method
- local truncation error, global error
- stability, consistence, convergence
- nonlinear equations
- singular perturbations and boundary layers
- high-order methods
- the five- and nine-point Laplacians


Part II. Numerical Methods for Time-Dependent ODEs.

- one-step methods
- Taylor series methods
- Runge-Kutta methods
- embedded methods and error estimation
- linear multistep methods
- predictor-corrector methods
- zero-stability and convergence
- absolute stability
- numerical methods for stiff ODEs
- differential-algebraic equations
- Hamiltonian systems and long time integration
- symplectic methods and their properties


Part III. Numerical Methods for Time-Dependent PDEs

- local truncation error and order of accuracy
- semi-discretization (method of lines): boundary conditions, stability 
and convergence
- fully-discrete schemes: general linear stability and convergence
- Lax equivalence theorem
- CFL condition
- modified equation

Stability for Constant Coefficient Problems

- Fourier analysis for scalar equations and for systems
- eigenvalue analysis

Variable Coefficient and Nonlinear Problems

- Freezing coefficients and dissipativity
- schemes for one-dimensional hyperbolic systems
- nonlinear stability and energy methods

Dispersion and Dissipation

- dispersion relation, phase velocity, group velocity
- the wave equation
- the KdV equation
- Lagrangian methods

Numerical Methods for Initial-Boundary Value Problems

- parabolic problems
- hyperbolic problems
- infinite or large domains and artificial boundaries

Several Space Variables and Splitting Methods

Discontinuities and Almost Discontinuities

- upwind and artificial viscosity
- one-dimensional scalar conservation laws
- strong stability preserving methods
- systems of conservation laws
- multidimensional problems