Which of the following statements is true?单项选择题

题目图片
A

a. Heun's method uses the gradient at the midpoint of the interval to represent the gradient of the entire interval.

B

b. The Euler method uses the gradients at the upper bounds of an interval to represent the gradient of the entire interval.

C

c. The Euler method uses the average of the gradients at the lower and upper bounds of an interval to represent the gradient of the entire interval.

D

d. Heun's method uses the average of the gradients at the lower and upper bounds of an interval to represent the gradient of the entire interval.

E

e. The midpoint method uses the average of the gradients at the lower and upper bounds of an interval to represent the gradient of the entire interval.

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Question textAnswer the following multiple choice questions about the theory behind numerically solving ODEs:1. What is the goal of numerically solving ODEs? Multiple choice 1 Question 1find a series of points that approximate [math: t(y)]find a series of points that approximate [math: y(t)]find a series of points that approximate [math: dy/dt]find a series of points that approximate [math: dt/dy]2. Which of the following statements is true regarding Euler's method in estimating [math: y(i+1)]?Multiple choice 2 Question 1It uses a single derivativeIt uses an average of two derivativesIt uses multiple derivativesIt uses no derivatives3. For a given ODE, what is the most practical way to minimise the error when using Euler’s method?Multiple choice 3 Question 1Ensure the second derivative of the RHS of the ODE is zeroUse a larger step sizeUse a smaller step sizeChoose an optimal starting point4. How do Heun's Method and the Midpoint Method improve on Euler's method?Multiple choice 4 Question 1Both of them use higher order derivativesBoth of them use the derivative at more than 2 pointsHeun's uses the derivative at the start and end of the step, while midpoint uses the derivative in the middle of the stepMidpoint uses the derivative at the start and end of the step, while Heun's uses the derivative in the middle of the stepCheck Question 1

Question textMatch the local and global truncation errors for the following ODE solving methods: Euler's method: Answer 1 Question 23[select: , local: O(h), global: O(h^2), local: O(h^2), global: O(h), local:(O(h^2), global: O(h^3), local: O(h^3), global: O(h^2), local: O(h^3), global: O(h^4)] Heun's method: Answer 2 Question 23[select: , local: O(h), global: O(h^2), local: O(h^2), global: O(h), local:(O(h^2), global: O(h^3), local: O(h^3), global: O(h^2), local: O(h^3), global: O(h^4)] Midpoint method: Answer 3 Question 23[select: , local: O(h), global: O(h^2), local: O(h^2), global: O(h), local:(O(h^2), global: O(h^3), local: O(h^3), global: O(h^2), local: O(h^3), global: O(h^4)]Check Question 23

Which method below uses an average slope of the start and end in an Euler-like step for its corrector step?

Which of the following method uses Euler's method as a predictor step with a half step size?

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