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
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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Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff464ebc2-eab2-44c1-923d-bd0dfc4fff5c%2F33182d2b-6c72-45dc-ab28-b4aa1cd65049%2F35czg7l_processed.png&w=3840&q=75)
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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