From Last Lecture: Singular Value Decomposition ODE-IVP: Numerical Integration Boundary Conditions: Initial Value Problems (IVPs) Boundary Conditions: Boundary Value Problems (BVPs) Integrals Can Be Rewritten as ODE-IVPs. Numerical Integration10.34, 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(λn,tk) = Σxi(tk)Ai(λn) + noise SVD ΣσiUi(tk)ViT(λn) 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 = mad2x/dt2 = F(x,dx/dt)/m dv/dt = F/m dx/dt = v()⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛vmvxFxVdtd, so setting x1=v and x2=x gives ()()xxmxFxxdtdF121=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛ g(x,dx/dt) = 0 DAE Î 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 Æ g(xi(t0),xj(tf)) = 0 “a mess!!” Integrals Can Be Rewritten as ODE-IVPs. I = 0)()(')'(00==∫tItfdtdIdttftt x = =⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛1)(1)(1/2xftfdtdIdtdtIxF(x) 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].⎟⎟⎠⎞⎜⎜⎝⎛==∫0010')'()(0txdttftxfttf Numerical Integration {ti} {f(ti)} t0 tff(t) I ~ Σf(ti)∆t rectangle method (not very effective) 10.34, Numerical Methods Applied to Chemical Engineering Lecture 12 Prof. William Green 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]. 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 Æ 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: N()NttfIIioldnewΔ+=Δ small ist if smallvery big 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® ª 4th order polynomial and 5th order polynomial * cut ∆t until 4th and 5th order extrapolations agree within certain tolerance10.34, Numerical Methods Applied to Chemical Engineering Lecture 12 Prof. William Green 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]. 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
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