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lecture 16 - CE 30125 Lecture 16 LECTURE 16 NUMERICAL...

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CE 30125 - Lecture 16 p. 16.1 LECTURE 16 NUMERICAL SOLUTION OF THE TRANSIENT DIFFUSION EQUATION US- ING THE FINITE DIFFERENCE (FD) METHOD Solve the p.d.e. Initial conditions (i.c.’s) Boundary conditions (b.c.’s) • Notes • We can also specify derivative b.c.’s but we must have at least one functional value b.c. for uniqueness. • This p.d.e. is classified as a parabolic type p.d.e. • The equation can represent heat conduction and mass diffusion. u t ----- D 2 u x 2 -------- = u x t = t o , ( ) u * o x ( ) = u x = x 1 t , ( ) u ** = 1 t ( ) u x = x 2 t , ( ) u ** 2 t ( ) =
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CE 30125 - Lecture 16 p. 16.2 Explicit Solution Procedure Evaluate the p.d.e. at point ( and ) Let’s use a forward difference approximation to evaluate at : accurate i j , ( ) i spatial index = j temporal index = t j+1 j j-1 2 1 j=0 (i,j) i=0 1 2 i-1 i i+1 n-1 n i=n+1 x Dt Dx known values unknown values u t ----- i j , u t ----- i j , u i j 1 + , u i j , Δ t ---------------------------- = O Δ t ( )
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CE 30125 - Lecture 16 p. 16.3 Use a central difference approximation to evaluate at : accurate Substituting into the p.d.e.: Solving for unknown represents the solution at the node and the time level. 2 u x 2 -------- i j , 2 u x 2 -------- i j , u i 1 j , + 2 u i j , u i 1 j , + Δ x ( ) 2 ----------------------------------------------------- = O Δ x ( ) 2 1 Δ t ----- u i j 1 + , u i j , ( ) D Δ x ( ) 2 ------------- u i 1 j , + 2 u i j , u i 1 j , + ( ) = u i j 1 + , i th j 1 + ( ) th u i j 1 + , Δ tD Δ x ( ) 2 ------------- u i 1 j , + 1 2 Δ tD Δ x ( ) 2 -------------- u i j , Δ tD Δ x ( ) 2 -------------- u i 1 j , + + =
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CE 30125 - Lecture 16 p. 16.4 Notes on solving the discrete approximations to the p.d.e. • Let’s examine the FD molecule: • One discrete equation can be written for each node (such that the number of unknowns equals the number of equations) • We can compute unknown nodal values of at the new time level directly from values of the previous time level (i.e. they are not coupled at the new time level). The order in which the computations are performed in space does not matter since the values at the new time level are entirely dependent on values at previous time levels. • Explicit Formula - one unknown pivotal (or nodal) value is directly expressed in terms of known pivotal values. j+1 j-1 j i-1 i i+1 known values unknown values u
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CE 30125 - Lecture 16 p. 16.5 Notes on time marching and accuracy • The process advancing from a known time level(s) to the unknown time level is called “time marching”.
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