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
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 discusse
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-Ni
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 wha
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 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
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 discuss
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 pro
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.
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
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
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