Lec19_Control_Volume_Time

Lec19_Control_Volume_Time - Lecture 19: Control-Volume...

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Lecture 19: ontrol olume Approach: Time ependent Solutions Control-Volume Approach: Time-Dependent Solutions
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Consistency, Stability and Convergence Numerical solution of PDE approaches the continuum solution as step sizes go to zero: convergent Individual terms approach continuum derivatives (Newtons’ limit to the derivative): consistent approximation Boundedness of solution: stability
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Von Neumann Stability Analysis for Model Equations For a stable solution, the errors must not amplify in time. That is: 1 1 n i n tability: Model Equation: i Stability: Diffusion restriction 2 2 CC D tx  t 2 1 2 Dt x Courant Condition 0 U  1 Ut x t The number is also called CFL (Courant-Fredericks-Lewy) number x   
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Explicit and Implicit Methods plicit Schemes: ompute the new Explicit Schemes: Compute the new concentrations on a cell by cell basis based on e concentrations at neighboring cells at the the concentrations at neighboring cells at the old time level plicit Scheme: ompute new Implicit Scheme: Compute new concentrations by inverting a matrix. erything is unknown until the matrix Everything is unknown until the matrix equation is solved for the new concentrations.
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Explicit Method 2 CC  Explicit Scheme 2 1 11 2 2 nn n n n ii i i i D tx C D x     FTCS Method  1 2 2 n n n i i i Dt C C C x    What is the condition for stability?
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This note was uploaded on 01/20/2011 for the course ENE 804 taught by Professor Hashsham,s during the Spring '08 term at Michigan State University.

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Lec19_Control_Volume_Time - Lecture 19: Control-Volume...

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