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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

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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.

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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.

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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

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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.

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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 92 235 (“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

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Simplest model of neutron diffusion Laplace operator: In spherical coordinates: So for spherically symmetric systems:

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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

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Consider a sphere of “25” where = mean time between fissions = avg. no. of neutrons produced per fission D = diffusion constant

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Separation of variables: an important technique where = effective neutron number Leads to

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Separation of variables: an important technique For sphere of radius R, can check solution With the boundary condition So critical mass is determined by

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Answers For Uranium: More accurate boundary condition gives 56 kg, and thick U tamper gives 15 kg

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Little Boy

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Chapter 9 Differential equations

Chapter 9 Differential equations

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