CS 205A class_17 notes

Scientific Computing

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CS205 – Class 17 Covered in class : All Reading : 9.3.9 1. Multivalue methods – efficiently use lower accuracy on higher derivatives a. Consider the Taylor expansion 2 13 () 2 nn n n tt t t x xt x x O t + Δ = + + Δ i.e. consider case where we have , t t x vx ao rv a == = . i. If n x has r Ot Δ errors than n t x can have 1 r Δ errors without ruining the accuracy, similarly n tt x can have 2 r Δ errors ii. e.g., 3 rd order accurate x G can be obtained with a 2 nd order accurate v G and 1 st order accurate a G iii. Solving x v G G as a standard system is overkill on v G b. Standard constant acceleration equations i. 2 1 2 n n t x v a + Δ =+ Δ+ GG G G quadratic position ii. 1 n vvt a + Δ G linear velocity iii. 1 aa + = constant acceleration (that is constant from time n to just before time n+1) 2. Newmark Method – most famous multivalue method in computational mechanics a. Actually a lot of methods in disguise b. 2 11 (1 2 ) 2 2 n n n t xxt v a a ββ ++ Δ ⎡⎤ + ⎣⎦ G G G c. (1 ) γγ Δ + d. Choice of , βγ parameters makes a specific method. i. 0 β γ - standard constant acceleration case (above) ii. 12 = , 1 = - piecewise constant, implicit acceleration 1. 2 2 n n t x v a Δ G G 2. n a Δ G 3. Second equation is the same as 1 st order accurate backward Euler 4. First equation is 1 1 2 vv + + + Δ G G which is the 2nd order accurate trapezoidal rule 5.
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

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CS 205A class_17 notes - CS205 Class 17 Covered in class:...

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