Review of accuracy analysis

1
Review of accuracy analysis
Euler Local error O(h2) Global error
O(h)
Runge-Kutta Order 4 Local error O(h5)
Global error O(h4)
But theres more to worry about stability
and convergence
2
Stability Convergence
Stability Suppose we perturb initial condition
by e. Then 1) effect ? 0, and 2) effect grows
only polynomially fast as a function of step size
h.
Convergence Solution of discrete problem ?
solution of continuous problem as h ? 0.
3
Stability Convergence
For Ordinary Differential Equations (ODEs),
Stability ? Convergence
But for Partial Differential Equations (PDEs),
where there are more than one variable--- time
and space, for example--- Stability and
convergence are not equivalent. We require an
additional condition.
4
Laxs Theorem
Consistent A finite-difference scheme is
consistent if the local truncation error ? 0 as
the grid size ? 0. (Not always true for PDEs, as
we shall see.)
Laxs Theorem If a finite-difference scheme for
an initial-value PDE is consistent, then
Stability ? Convergence
5
PDEs
Partial differential equations are at the very
heart of many sciences, and provide our best
understanding of the way the world works. Some
examples
Quantum mechanics propagation of waves of all
kinds elasticity diffusion of
particles, population, prices, information
spread of heat electrostatic field magnetic
fields, fluid flow etc., etc., etc.
6
To see the power
Suppose you work for the government and your job
is to worry about the possibility of terrorist
nuclear weapons. What is the critical mass of
U92235 (25)?
The following material was classified, but is now
public see The Los Alamos Primer The first
lectures on how to build an atomic bomb, R.
Serber, Univ. of Calif. Press, Berkeley, 1992.
QC773 .A1S47
7
(No Transcript)
8
Simplest model of neutron diffusion
Laplace operator
In cylindrical coordinates
So for spherically symmetric systems
9
Consider a sphere of 25
Let N(t,x,y,z) be the number of neutrons in a
tiny cube and consider the net growth of N at any
given point in space and any particular time
Rate of change of neutron flux
Diffusion influx
fission
10
Consider a sphere of 25
where mean time between fissions
avg. no. of neutrons
produced per fission D
diffusion constant
11
(No Transcript)
12
Separation of variables an important technique
where effective neutron number
Leads to
13
Separation of variables an important technique
For sphere of radius R, can check solution
With the boundary condition
So critical mass is determined by
14
Answers
For Uranium
More accurate boundary condition gives 56 kg,
and thick U tamper gives 15 kg
15
Little Boy