10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 33 Operator Splitting Strang Splitting Problem Set 11 t e ln rand t rand 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 t failure failure ln rand The failure is from the continuum equations Operator Splitting Problem A and B have no time derivative y A y B y y t 0 y t 0 t t 1 Split Operators y A y t y B y t y A y t y t 0 y 0 t0 t0 t 2 y y t 0 y t 0 t 0 t y t y t 0 y 2 t t 0 t 0 t 2 y t t Error O t 2 0 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 that are tailored for each part Generalization y n xi y x i t S y x i t t 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 The first term on the right hand side is spatially non local usually not that stiff The stiffness in the first term comes from the use of a fine mesh The second term on the right hand side is spatially local and stiff it depends on all species Most splitting methods give error O t Strang splitting is better because it has an error O J t 2 So we say Strang splitting is time accurate The error is difficult to bound Usually one runs the solution with a different t Definition of A and B Nmesh Nspecies number of y s 2 2 Direct coupled Number of variables scales as O N mesh N sp This leads to a storage problem y Sy t 2 Nspecies at local point J Nsp2 The number of variables scales as O N mesh N sp Can then solve each mesh point in parallel Steady State Considerations and Time March If you only care about steady state F y 0 the usual approach is to use a Newton type solver J y F y guess At t Implicit Euler gives the same result With a reacting flow a good guess is difficult Possible approaches are Work on finding a good guess Solve a simpler problem then change the problem gradually into the one you have Time march to steady state see below this works for continuous stirred tank reactors CSTRs y F y t One may have to march for a while y t 0 y 0 t y t l arg e y guess At large t the solution may approach a value that can be used as the initial guess for Newton s method In biology there are often multiple steady states so the solution might jump from one state to another In laminar flow one can analyze the steady states In turbulent flow one can calculate time average values but not fluctuations Finding steady state can be difficult if the problem has more than 10 variables use time march Explicit Euler Recall that Explicit Euler is numerically unstable y t t y t F y t t 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 33 Page 2 of 3 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 y new y old F y old t Time Marching With Implicit Euler Although Implicit Euler is not time accurate it is unconditionally numerically stable y t t y t F y t t t y new y old F y new t F y new y old F y new t J mn Fm y guess 2 y nnew solves implicit Euler mn J y guess 2 t J mn y refined F y guess2 I t J y refined y guess 2 F y guess 2 t Right hand side is close to zero t F y t For small t I t J I which is well conditioned Time march with implicit Euler instead of Newton s method Implicit Euler allows large time steps 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 33 Page 3 of 3 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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