PHYS 7412
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
Lecture 22
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
Lecture 19
Partial differential equations
The Schrdinger equation
More than 1+1.
Operator splitting.
Elliptic equations: Fourier methods.
Elliptic equations: relaxation methods.
The Schrdinger equation:
We have driven home the message that accuracy is not
Lecture 18
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:
The met
Lecture 17
Partial differential equations.
Flux-conservative equations, FTCS scheme.
Von Neumann stability analysis.
Lax scheme.
Other kinds of errors.
This is a vast subject. Dealing with PDEs is what several physicists
do for a living (including your in
Lecture 16
Two point-boundary value problems
Rayleigh-Ritz method.
Integral equations
General definitions
Volterra equations
Singular kernels
Rayleigh-Ritz method:
Up to now our approach to both differentiation and integration has been
to use finite diffe
Lecture 15
Two point boundary value problems
Shooting method
Relaxation methods.
Adaptive meshes.
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
Lecture 12
Eigensystems: conclusion
Ordinary differential equations:
Initial and boundary value problems
The Euler method
The Runge-Kutta method.
Stepsize control.
The QL decomposition we discussed last class is too costly
computationally for the alg
Lecture 8
Random numbers
Non-uniform probability distributions
The Metropolis algorithm
Random numbers:
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
Polynomial interpolation
Rational approximation
Coefficients of the polynomial
Interpolation:
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
c
Lecture 7
Pad approximation.
Monte Carlo method of integration.
Pad approximations:
A quotient of two polynomials:
Is said to be a Pad approximation to
a power series
f ( x ) = ck x k
k =0
If f(0)=R(0) and,
That is, the ratio R(x) has a power series exp
Lecture 13
Ordinary differential equations
Embedded Runge-Kutta
Modified midpoint method.
Bulirsch-Stoer method.
Predictor-corrector methods.
Embedded Runge-Kutta:
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
answer.
Note:
1. Use the following values to construct a