lec33_12042006 - 10.34, Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #33: Operator Splitting. Strang Splitting. Problem Set 11 () rand t ln = τ rand e t Poisson Statistics Poisson statistics can be found in applications such as radioactive decay, droplets hitting a roof, and average time between uncorrelated events. Gillespie Algorithm – Must run many times Continuum view: assume Poisson statistics ( rand t failure failure ln ) = The τ failure is from the continuum equations. Operator Splitting Problem A ˆ and B ˆ have no time derivative. ()( y B y A t y ˆ ˆ + = ) () ( ) ±² ±³ ´ 1 0 0 t t y t y Δ + Split Operators ( y A t y ˆ = ) () 0 0 y t y = 2 0 0 t t t Δ + y ( y B t y ˆ = ) () = y t y 0 t t t Δ + 0 0 y ( y A t y ˆ = ) = Δ + y t t y 2 0 t t t t Δ + Δ + 0 0 2 ( ) ±² ±³ ´ 2 0 t t y Δ + Error O(( Δ t) 2 ) Are 1 and 2 the same? No. They are not the same. Operator splitting introduces error. Why do we do this? In a reacting flow problem, maybe the two parts have solution methods
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lec33_12042006 - 10.34, Numerical Methods Applied to...

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