Math 552 Course Schedule

Content
 Info Schedule Homework Hints Q & A Programs Software

MATH 552 Numerical Solution of PDEs

Spring 2013 Course Outline and Tentative schedule
Many well-known and widely used mathematical models of interesting and important scientific and engineering problems are formulated as partial differential equations. These models, due to their enormous complexity and possible singularity in their solutions, are often challenging problems to be solved numerically. This course will address both the analytical and the computational aspects of the subject. While offering an overview to various popular methods and well established theory, the students will also get a hands-on experience in through the solution of some models problems by themselves.
A tentative outline is as follows:
• Introduction to the course and review of PDEs (1 lecture)
• Basic methods for numerical PDEs:
• an illustration via a model Poisson Equation (5 lectures)
Finite difference method
Finite element method
Finite volume method
Spectral method
Approximation and convergence properties
• More on elliptic equations in general domain (5 weeks)
Fundamental theory, maximal principle,
Variational principle, regularity
Structured and unstructured meshing
Basic finite difference schemes, boundary treatments
Discrete maximal principle and M-matrices
Convergence and Error estimates
Finite element spaces and basic error estimates
Conforming finite element method,
More general finite element methods
Finite volume and co-volume methods
Problems in exterior and infinite domain
Artificial boundary conditions
Boundary integral and boundary element methods
Methods for solving the discrete systems
Domain decomposition and parallel algorithms
Nonlinear variational problems
• Parabolic problems (2.5 weeks)
Methods of lines
Finite difference methods, Stability analysis
Fully discrete schemes,
Alternating and splitting methods,
Explicit-Implicit schemes
• Hyperbolic/convection dominated problems (2.5 weeks)
Basic difference approximations,
Upwinding, Dissipation and Dispersion,
Stability analysis,
Methods for scalar 1d conservation laws, Shocks
Methods foe Hamilton-Jacobi equation
• Schrodinger equation (1 week)
Splitting methods