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 as ε 0, and 2) effect grows only polynomially fast as h 0. 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: In spherical coordinates: So for spherically symmetric systems:
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
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 where = effective neutron number Leads to
Separation of variables: an important technique For sphere of radius R, can check solution With the boundary condition So critical mass is determined by
Answers For Uranium: More accurate boundary condition gives 56 kg, and thick U tamper gives 15 kg