heat - estimate u j +1 i by approximating the derivatives u...

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Math 473/Numerical Analysis/Fall 2009 The Heat Equation The heat equation is the partial differential equation u t = u xx . Solv- ing this equation analytically is beyond the scope of the course. We instead solve it numerically. A solution of the heat equation is in principle a function u ( x,t ). We will produce estimates for this function at points of a grid. Choose a grid mesh Δ x for the x direction, and Δ t . in the t direction. Let u j i be an estimate for u ( i Δ x,j Δ t ). If we know u j i - 1 , u j i and u j i +1 , we can
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Unformatted text preview: estimate u j +1 i by approximating the derivatives u t and u xx at the point u ( i Δ x,j Δ t ): u xx ’ u j i-1-2 u j i + u j i +1 (Δ x ) 2 u t ’ u j +1 i-u j i Δ t By the heat equation, these are equal. The only term we do not know is u j +1 i , so we can make it the subject of the equation. Letting s = Δ t/ (Δ x ) 2 , this gives u j +1 i = su j i-1 + (1-2 s ) u j i + su j i +1 . For stability we need s < 1 2 ....
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This note was uploaded on 10/10/2010 for the course MATH 01:640:111 taught by Professor Carey during the Spring '10 term at Saint Louis.

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