Introduction to the implementation and analysis of numerical algorithms for the numerical solution of the classic partial differential equations of science and engineering.
- Quick Review of Basic Models - Elliptic, parabolic, and hyperbolic problems, classification and behavior of solutions, numerical concerns
- Elliptic Problems and the Finite Element Method - Two-point boundary value problems, Laplace and Poisson equations, variational formulation, finite element methods, interpolation theory, quadrature, energy norm, a priori convergence, order of convergence
- Brief Review of Numerical Linear Algebra - Specialized to systems arising from discretization of differential equations: sparse and banded matrices, direct methods, basic iterative methods
- Parabolic Problems and the Method of Lines - Explicit and implicit discretization schemes, numerical stability, stiffness and dissipativity, convergence
- Hyperbolic Problems - Transport equation, characteristics, finite difference schemes, stability, dissipativity, dispersion, the CFL condition, convergence, the wave equation
- Miscellaneous Topics as Time Permits - Error estimation, adaptive schemes, conservation laws