lec09_09272006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #9: Function Space vs. Real Space Methods for Partial Differential Equations (PDEs). Practical Utility of Eigen-Analysis Stability of Steady States Stability of Differential Equations Bifurcations Correlated Uncertainties Function Spaces vs. Real Space F (x ) = 0 Broyden Conservation Laws Reasonableness Check Newton kinetic equations Î x soln Does solution make sense? Trust-Region Equations *physically * m a t h e m a t i c a l l y If x soln is unstable, then in reality x soln is useless. *you converged to the wrong solution Stability soln soln soln soln 0 ( ) is the origin of equation ( ) ( ) ( ) ( )( ) ...... () *neglected higher order terms d dt d dt dd dt dt == + + = x Fx x Fx Fx Jx x x xx x Y o u w a n t e i g ( J ) all < 0 If this condition is true, you will find x soln I f e i g ( J ) is positive: unstable state, will go farther away from solution with time. It is always important to find the Jacobian. If the eigenvalue of the Jacobian is less than zero, you have found the stable steady state . Test to tell if you got the correct solution ) ( x F x = dt d x = x o w = x (t) – x ref (t) Figure 1. Stable and unstable solutions. ) ( x F x = dt d x = x ref + w Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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dt d dt d dt d ref w x x + = =F (x ref + w ) = F (x ref ) + J 10.34, Numerical Methods Applied to Chemical Engineering Lecture 9 Prof. William Green Page 2 of 5 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.

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lec09_09272006 - 10.34, Numerical Methods Applied to...

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