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Chaos

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Chaos Powered By Docstoc
					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 there’s more to worry about:
    stability and convergence
    Stability & Convergence
Stability: Suppose we perturb initial
condition by ε. 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.
     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.
          Lax’s 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.)
Lax’s Theorem: If a finite-difference
scheme for an initial-value PDE is
consistent, then
       Stability ↔ Convergence
                   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.
       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
Simplest model of neutron diffusion
Laplace operator:
                       2 f 2 f 2 f
     2 f ( x, y, z )  2  2  2
                       x   y   z
In cylindrical coordinates:
          1  2 f
     f  2 (r
     2
                   )  terms in  ,
         r r   r
So for spherically symmetric systems:
               1  2 f
          f  2 (r
           2
                        )
              r r   r
   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:

      N           1
          D N 
             2
                       N
      t           
Rate of change of   Diffusion influx   fission
neutron flux
 Consider a sphere of “25”

where    = mean time between fissions
         = avg. no. of neutrons
            produced per fission
        D = diffusion constant
 Separation of variables: an
    important technique
                         t / 
     N  N1 ( x, y, z )e
where   = effective neutron number

Leads to
                 1  
       N1  
        2
                          N1
                  D
    Separation of variables: an
       important technique
For sphere of radius R, can check solution
                  sin r / R
             N1 
                       r
With the boundary condition
                            D
                            2
                1         2
                            R
So critical mass is determined by
                   D2
              R 
                2

                   1
              Answers
   For Uranium:
           2 D  220
            2.3
          Rcritical  13.5 cm
          Masscritical  200 kg

More accurate boundary condition gives
56 kg, and thick U tamper gives 15 kg
Little Boy

				
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