10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 12 Ordinary Differential Equation Initial Value Problems ODE IVPs and Numerical Integration From Last Lecture Singular Value Decomposition Sobserved n tk xi tk Ai n noise SVD iUi tk ViT n Fixed in time Amplitude changes with wavelength For a system where only one chemical absorbs light expect one singular value to be bigger The rest of the singular values relate to noise Look in Beer s textbook for exact notation ODE IVP Numerical Integration dx dt F x ODE F ma d2x dt2 F x dx dt m dv dt F m dx dt v d V F x v d x1 F x m F x m so setting x1 v and x2 x gives dt x dt x 2 x1 v g x dx dt 0 DAE differential algebraic equations Boundary Conditions Initial Value Problems IVPs x t0 x0 As you move away from x0 you are not sure whether you are correct That is why the first step if incorrect can send you far away from the answer and errors can multiply Boundary Conditions Boundary Value Problems BVPs x t0 xi0 xj tf xjf g xi t0 xj tf 0 a mess Integrals Can Be Rewritten as ODE IVPs I t t0 f t dt I x t dI f t dt I t 0 0 dx dI dt f t f x 2 F x dt 1 1 1 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 tf x1 t f f t dt t0 0 x0 t0 Numerical Integration ti f ti I f ti t f t rectangle method not very effective t0 tf Figure 1 Graph of a function over a time interval Trapezoid Rule in MatLAB trapz Newton Cotes Rules polynomial interpolation between f ti Simpsons Rule a k a Crafty Rule f t a0 a1t a2t2 Error O t3 area under parabolas This technique involves 2 times more work but is many times more accurate 1 High order methods have much faster convergence as t small If too high order we are fitting the noise to f t but if too small t we are adding big numbers and small numbers I new I old N big f t t Ni very small if t is small 2 Adaptive step sizing adaptive meshing With adaptive step sizing or adaptive meshing the time step sizes are smaller close to the steep peak and larger along the tail where the function is not changing as quickly t0 tf ti Figure 2 Graph of a function with a steep peak ODE45 in MATLAB 4th order polynomial and 5th order polynomial cut t until 4th and 5th order extrapolations agree within certain tolerance 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 12 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 Runge Kutta 4 5 method x t1 x t0 t f x0 O tN x t2 x t1 t f x1 O tN No way to guarantee global error With ODE s the error in each step accumulates in a nonlinear way In integration the errors are present for each step separately additive Computing the integral incorrectly for a previous step does not affect the step at hand 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 12 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
View Full Document
Unlocking...