Lecture 1: Errors
Internal representation of numbers
Round-off and Truncation errors
No matter how sophisticated modern computers appear to be, at their
very essence they are machines that add and subtract numbers.
These numbers are internally
The finite element method
The finite element method:
The finite element method for elliptic PDEs is the generalization to
higher dimensions of the Rayleigh-Ritz type of methods we discussed
for two-point boundary value problems.
The details ho
Partial differential equations
The Schrdinger equation
More than 1+1.
Elliptic equations: Fourier methods.
Elliptic equations: relaxation methods.
The Schrdinger equation:
We have driven home the message that accuracy is not
Partial differential equations:
Schemes that are second order in time:
Staggered leapfrog and Lax-Wendroff.
Shocks in fluids.
Diffusive initial value problem: fully
implicit and Crank-Nicholson methods.
Second order accuracy in time:
Partial differential equations.
Flux-conservative equations, FTCS scheme.
Von Neumann stability analysis.
Other kinds of errors.
This is a vast subject. Dealing with PDEs is what several physicists
do for a living (including your in
Two point-boundary value problems
Up to now our approach to both differentiation and integration has been
to use finite diffe
Two point boundary value problems
Singularities in the domain.
The shooting method:
Let us assume that at point x1 one has to specify N boundary values
for the ODE, corresponding to the initi
Ordinary differential equations:
Initial and boundary value problems
The Euler method
The Runge-Kutta method.
The QL decomposition we discussed last class is too costly
computationally for the alg
Non-uniform probability distributions
The Metropolis algorithm
It might appear strange to ask a computer, which carries out
deterministic procedures, to produce a random result. In fact, it
simply cannot. Some p
Lecture 5: Interpolation
Coefficients of the polynomial
Sometime we know the values of a function f(x) for a finite set
of points xi. Yet we want to evaluate f(x) for other values perhaps
Monte Carlo method of integration.
A quotient of two polynomials:
Is said to be a Pad approximation to
a power series
f ( x ) = ck x k
If f(0)=R(0) and,
That is, the ratio R(x) has a power series exp
Ordinary differential equations
Modified midpoint method.
The Runge-Kutta formulas with M>4 require more than M evaluations
of the function. This in
Physics 7412, Fall 2013
Problem set 2
Due date: 9/25/13
These problems can be addressed with a hand-held calculator, Mathematica, Maple, etc. Either that or writing a computer code is acceptable as
1. Use the following values to construct a