Consider the Initial Value Problem du/dt = 50/u-50u, 0...
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Consider the Initial Value Problem du/dt = 50/u-50u, 0<t<1, u(0) = √2. We will use Implicit Euler and Implicit Trapezoid to solve this nonlin- ear problem. (a) First, write out each of the schemes for this problem (the equation relating ui+1 to uż). Formulate each as an equation of the form g(i+1, Ui) = 0, a root finding problem. (b) In this particular case, one could solve the root finding problems analytically. However, we will not do so. Instead, we will use Newton's method. As always, this is done by making an initial ,(0) guess for what ₁+1 might be, call this u, then generating a (k) sequence of successive guesses ut based on the standard New- ton's method formula. For each of the two schemes, write out ,,(k+1) the equation for generating the next guess ut from the prior (k) guess u via Newton's method. (c) Now, implement the above in code, and solve the system with each method using N = 10 equal width subdivisions of the inter- val 0 < t < 1. For your root finding, use a stopping tolerance of € = 10-6. For your initial Newton's method guess u at each ,(0) timestep, use the solution you found for the prior timestep, ₁. Plot the two solutions. (d) The exact solution to this problem is u(t)=√1+e-100€. Given this and your numerical solutions above, plot the error E₁ |u(ti) - ui as a function of t for each method, and comment on what you find. = Consider the Initial Value Problem du/dt = 50/u-50u, 0<t<1, u(0) = √2. We will use Implicit Euler and Implicit Trapezoid to solve this nonlin- ear problem. (a) First, write out each of the schemes for this problem (the equation relating ui+1 to uż). Formulate each as an equation of the form g(i+1, Ui) = 0, a root finding problem. (b) In this particular case, one could solve the root finding problems analytically. However, we will not do so. Instead, we will use Newton's method. As always, this is done by making an initial ,(0) guess for what ₁+1 might be, call this u, then generating a (k) sequence of successive guesses ut based on the standard New- ton's method formula. For each of the two schemes, write out ,,(k+1) the equation for generating the next guess ut from the prior (k) guess u via Newton's method. (c) Now, implement the above in code, and solve the system with each method using N = 10 equal width subdivisions of the inter- val 0 < t < 1. For your root finding, use a stopping tolerance of € = 10-6. For your initial Newton's method guess u at each ,(0) timestep, use the solution you found for the prior timestep, ₁. Plot the two solutions. (d) The exact solution to this problem is u(t)=√1+e-100€. Given this and your numerical solutions above, plot the error E₁ |u(ti) - ui as a function of t for each method, and comment on what you find. =
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Related Book For
Calculus For Scientists And Engineers Early Transcendentals
ISBN: 9780321849212
1st Edition
Authors: William L Briggs, Bernard Gillett, Bill L Briggs, Lyle Cochran
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