Course Outline

The following topics are available:

Partial Differential Equations

As time and interest permits. Systems of first order equations arising in heat transfer, fluid and solid mechanics. Transformation of second order equations to first order systems. Quasi-linear and linear systems. Conservation laws. Shocks. Classification of partial differential equations and systems. Hyperbolic equations: characteristics, domains of influence and dependence, properly posed initial-boundary value problems. Fourier analysis. Parabolic equations, Fourier solutions, initial and boundary conditions. Elliptic equations, boundary conditions, Fourier solution, properly posed problems. [3,6,13,9]

Finite Difference Methods in One Dimension

As time and interest permits. Finite difference discretizations. Physical justifications. CFL condition, domain of dependence, maximum principles. Fourier solutions of difference equations. Taylor series expansions. Consistency, stability, dissipation and dispersion. Example problems from solid and fluid mechanics and heat transfer. Explicit and implicit schemes, upwinding, monotonicity preservation. [6,13,9,5]

Discretization Methods in Multiple Dimensions

As time and interest permits. General principles of curvi-linear grid generation. Finite difference discretizations on these grids. Finite volume discretizations. Galerkin and sub-domain finite element disretizations. [5]

Finite Element Methods in One Dimension

As time and interest permits. Weak formulation, Galerkin, collocation. Rayleigh-Ritz formulation. Finite elements, Lebesque integration, completeness, reduction of order. Energy/extremum principles. Convergence in natural and standard norms. [12,10].

Solution Methods in Multiple Dimensions

As time and interest permits. Discussion of direct and iterative solution procedures. Approximate factorization techniques, ADI and fractional step methods. Jacobi, Gauss-Seidel, SOR, multigrid, conjugate gradients, ILU methods. [13,9,4]

Spectral Methods

As time and interest permits. Introduction to the fast Fourier transform. Spectral accuracy. Spectral Galerkin, Tau, collocation. [1,14]