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i) Derive an explicit and an implicit multistep methods that use three nodes tn, tn-1 and tn-2 and determine their truncation errors. Are the methods consistent? Explain. ii) Determine whether the methods are convergent and find and plot the regions of ab- solute stability for the schemes. iii) Implement the derived methods and Euler's method and use them to solve the fol- lowing two initial value problems. (a) y = −(y+1)(y+3) with y(0) = -2 on t = [0,2] (exact solution y(t) = −3+ (22-3+1 (b) y=y/t(y/t)² with y(1) = 1 on t = [1,2] (exact solution y(t)=1+Int). iv) Does the global error for each of the methods behave as expected when h is halved? Use the technique involving taking logarithms of the error discussed in class and produce order graphs that show the three methods converge with the expected order. v) Produce a precision diagram which includes all three multistep methods (don't include divergent methods in the precision diagrams but demonstrate that they diverge on a different graph). Using the precision diagram discuss comparative performance of the methods.

i) Derive an explicit and an implicit multistep methods that use three nodes tn, tn-1 and tn-2 and determine their truncation errors. Are the methods consistent? Explain. ii) Determine whether the methods are convergent and find and plot the regions of ab- solute stability for the schemes. iii) Implement the derived methods and Euler's method and use them to solve the fol- lowing two initial value problems. (a) y = −(y+1)(y+3) with y(0) = -2 on t = [0,2] (exact solution y(t) = −3+ (22-3+1 (b) y=y/t(y/t)² with y(1) = 1 on t = [1,2] (exact solution y(t)=1+Int). iv) Does the global error for each of the methods behave as expected when h is halved? Use the technique involving taking logarithms of the error discussed in class and produce order graphs that show the three methods converge with the expected order. v) Produce a precision diagram which includes all three multistep methods (don't include divergent methods in the precision diagrams but demonstrate that they diverge on a different graph). Using the precision diagram discuss comparative performance of the methods.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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i) Derive an explicit and an implicit multistep methods that use three nodes tn, tn-1 and tn-2 and determine their truncation errors. Are the methods consistent? Explain. ii) Determine whether the methods are convergent and find and plot the regions of ab- solute stability for the schemes. iii) Implement the derived methods and Euler's method and use them to solve the fol- lowing two initial value problems. (a) y = −(y+1)(y+3) with y(0) = -2 on t = [0,2] (exact solution y(t) = −3+ (22-3+1 (b) y=y/t(y/t)² with y(1) = 1 on t = [1,2] (exact solution y(t)=1+Int). iv) Does the global error for each of the methods behave as expected when h is halved? Use the technique involving taking logarithms of the error discussed in class and produce order graphs that show the three methods converge with the expected order. v) Produce a precision diagram which includes all three multistep methods (don't include divergent methods in the precision diagrams but demonstrate that they diverge on a different graph). Using the precision diagram discuss comparative performance of the methods.
Transcribed Image Text:i) Derive an explicit and an implicit multistep methods that use three nodes tn, tn-1 and tn-2 and determine their truncation errors. Are the methods consistent? Explain. ii) Determine whether the methods are convergent and find and plot the regions of ab- solute stability for the schemes. iii) Implement the derived methods and Euler's method and use them to solve the fol- lowing two initial value problems. (a) y = −(y+1)(y+3) with y(0) = -2 on t = [0,2] (exact solution y(t) = −3+ (22-3+1 (b) y=y/t(y/t)² with y(1) = 1 on t = [1,2] (exact solution y(t)=1+Int). iv) Does the global error for each of the methods behave as expected when h is halved? Use the technique involving taking logarithms of the error discussed in class and produce order graphs that show the three methods converge with the expected order. v) Produce a precision diagram which includes all three multistep methods (don't include divergent methods in the precision diagrams but demonstrate that they diverge on a different graph). Using the precision diagram discuss comparative performance of the methods.
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