Math 572 - Spring 2008 - Course Outline/Diary
Material will be entered here in reverse order, so future and current
information will be at the top. I'll attempt to document what
happened in class with references to the text, handouts, etc.
- Thursday, 2/7: Switch to two dimensions: basic formulas,
systems, ordering unknowns, etc.
- Tuesday, 2/5: More topics: nonlinear BVPs, nonuniform grids,
adaptive meshes, general linear BVPs. Sample code:
bvp.m (Uniform Mesh)
bvp2.m (Non-Uniform Mesh)
- Thursday, 1/31: Examples and extensions to nonlinear BVPs
- Tuesday, 1/29: Stability, Consistency and Convergence (the big
result for solving DEs)
- Thursday, 1/24: The third approach to finite difference formulas is to
use polynomial interpolant. This is a good way to derive the formulas.
Also, we set up our first finite difference method problem for
u'' = f with u(0) = a, u(L) = b. Replacing u'' by a finite difference
formula and then replacing u(x) by appropriate U_i, we have a linear
system to solve. Next we need to talk about errors.
- Tuesday, 1/22: Accuracy of finite difference formulas can be
tested/determined by applying to polynomials as there is a relation
between the accuracy (the power of h), the order of the derivative
in the leading error term, and thus its effect on polynomials.
- Thursday, 1/17: Norms: Vector, Function, Grid Function (A.3,4,5); Finite
Differences (Chapter 1) (finally!) Unless otherwise stated every norm
will be a function or a grid function norm and the context should tell
you which of those two. For finite differences: get used to Taylor expansions
(Homework #1 Due Thurs 1/24)
- Tuesday, 1/15: More basics:
Calculus Basics: Taylor's Theorem in 1 and multi-dimensions: the main
tool for analyzing methods for
(Summary), Errors: Big Oh notation
(A.2). If there's an important theme in these, it's that
theory and/or mathematically
accurate statements lead directly to results which are computable. In
the case of Taylor, we get a polynomial and (with a bound on the derivative)
a simple expression for the error. With Big Oh we have a precise statement
of how an expression depends on a parameter (h), but can also suppose
such a relation and by computing certain values, determine the exact
nature of that relationship.
- Thursday, 1/10: Handout syllabus, go
over course structure, goals, work, etc. Begin with the basics:
Differential Equations (E.1): PDE vs. ODE, Initial Value vs. Boundary
Value, the canonical problems: Poisson, Heat and Wave Equations for
PDEs, link between the problems and the structure of the course: Poisson and
BVPs, then ODEs (as IVPs), then link together for heat and wave-type equations.
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