Lecture16wn_NucE309FA11

Lecture16wn_NucE309FA11 - NucE 309 Fall 2011 Lecture 16...

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16.1 NucE 309, Fall 2011, Lecture 16 19.5 Numerical Integration: Pages 817-818 (was 17.5 pages 869-870) We would like to approximate the value of the integral () b a Jf xd x by dividing the range into segments and approximating f(x) dx over each segment. Divide the range into n segments. The width of each segment is: ba h n A. Rectangular Rule. Approximate f(x) dx using the value of f(x) at the midpoint of each segment.  1/2 3/2 5/2 ( ) ( ) ( ) ( )  b n a J fxd x hfx fx B. Trapezoidal Rule. Approximate using the area of a trapezoid between x i and x i+1 . The area of that trapezoid is 1 ( ) 2  ii h dA f x f x . Add up all of these dA to give   11 2 2 3 3 2 2 1 1 () () ( ) ( ) ( ) () 2   b a nn n x d x h f af x f x f x f x f x f x f x f x f x f x f b simplifies to: 12 22 b a J f x d x hf x f x f b
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16.2 19.5 Introduction to Finite Difference Methods: Pages 827-828 (was 17.5 pages 879-880) Finite differences are approximations to differential equations using a discretization of the continuous domain. Usually define locations on a grid pattern where velocities, pressure, and temperature will be determined. We will look at a computational grid of points which are equally spaced in x and equally spaced in y . The spacing x will be different than y .
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This document was uploaded on 10/27/2011 for the course NUC E 309 at Penn State.

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Lecture16wn_NucE309FA11 - NucE 309 Fall 2011 Lecture 16...

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