Review of accuracy analysis Euler: Local error = O(h 2 ) Global error = O(h) Runge-Kutta Order 4: Local error = O(h 5 ) Global error = O(h 4 ) 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 U (“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, 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 influxfission

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

Little Boy