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(2) You've used a numerical method with second-order global error to estimate the value of a solution of a differential equation, and using 4 time steps you've got an error not exceeding 0.25. How many times must you halve the step size to be sure the error is less than 0.0001? How does the answer change if the method were fifth-order instead?

(2) You've used a numerical method with second-order global error to estimate the value of a solution of a differential equation, and using 4 time steps you've got an error not exceeding 0.25. How many times must you halve the step size to be sure the error is less than 0.0001? How does the answer change if the method were fifth-order instead?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 1CR
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(2) You've used a numerical method with second-order global error to estimate the value of a solution of a differential equation, and using 4 time steps you've got an error not exceeding 0.25. How many times must you halve the step size to be sure the error is less than 0.0001? How does the answer change if the method were fifth-order instead?
Transcribed Image Text:(2) You've used a numerical method with second-order global error to estimate the value of a solution of a differential equation, and using 4 time steps you've got an error not exceeding 0.25. How many times must you halve the step size to be sure the error is less than 0.0001? How does the answer change if the method were fifth-order instead?
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If the numerical method used has a second-order global error, we know that the error decreases quadratically with the step size, h. Therefore, if we halve the step size, the error should decrease by a factor of 4.

Let E(h) be the error associated with a step size h. If using a step size h results in an error not exceeding 0.25, then we have E(h) ≤ 0.25.

 

 

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