Multi-Step Methods in Numerical Analysis: Learn and Master Advanced Mathematics

Question 1

What is a defining characteristic of multi-step methods in numerical analysis?

Accepted Answer

They are more accurate than single-step methods.

Answer

They only use the current solution value in calculations.

Answer

They are always stable.

Answer

They require less computation than single-step methods.

Question 2

Which of the following methods is **not** a multi-step method for solving ordinary differential equations?

Accepted Answer

Euler's method

Answer

Adams-Bashforth method

Answer

Adams-Moulton method

Question 3

How does stability analysis assist in assessing the accuracy of numerical solutions obtained using multi-step methods?

Accepted Answer

It determines the maximum step size for a stable and reliable numerical solution.

Answer

It ensures convergence of the numerical solution to the exact solution as the step size approaches zero.

Answer

It classifies the multi-step method as explicit or implicit.

Answer

It suggests that stability analysis is unnecessary for accurate multi-step methods with high order of accuracy.

Question 4

Which of the following is a multi-step method used for solving ordinary differential equations (ODEs)?

Accepted Answer

Adams-Bashforth method

Answer

Euler method

Answer

Lagrange method

Answer

Runge-Kutta method

Question 5

What is the key difference between the Adams-Bashforth and Adams-Moulton methods?

Accepted Answer

Adams-Bashforth is an explicit method, while Adams-Moulton is an implicit method.

Answer

Adams-Bashforth is a one-step method, while Adams-Moulton is a multi-step method.

Answer

Adams-Bashforth is only suitable for stiff equations, while Adams-Moulton is only suitable for non-stiff equations.

Question 6

Which technique is commonly employed to estimate the error in a multi-step method?

Accepted Answer

Richardson extrapolation

Answer

Analytic solution

Answer

Collocation method

Answer

Euler method

Question 7

What is the order of accuracy of the Adams-Bashforth 3 method?

Accepted Answer

3

Answer

4

Answer

2

Answer

1

Question 8

Which of the following is a multi-step method for solving ordinary differential equations?

Accepted Answer

Adams-Bashforth method

Answer

Runge-Kutta method

Answer

Euler method

Answer

Symplectic Euler method

Question 9

What is the order of accuracy of the Adams-Moulton method?

Accepted Answer

p+1

Answer

p

Answer

p-1

Question 10

Which of the following is a technique used to estimate the error in a multi-step method?

Accepted Answer

Richardson extrapolation

Answer

Finite difference method

Answer

Taylor series

Question 11

What is the main disadvantage of multi-step methods compared to single-step methods?

Accepted Answer

Starter values are required

Answer

Higher computational cost

Answer

Reduced accuracy

Question 12

What is the relationship between the local truncation error and the global truncation error in a multi-step method?

Accepted Answer

The global truncation error is the sum of the local truncation errors

Answer

The global truncation error is the product of the local truncation errors

Answer

The local truncation error is the derivative of the global truncation error

Question 13

The general formula for the Adams-Bashforth method is:

Accepted Answer

y_n+k = y_n + h(b_0f_n + b_1f_n-1 + ... + b_kf_n-k)

Answer

y_n+k = y_n + h*f_n

Answer

y_n+k = y_n + h(a_0f_n + a_1f_n-1 + ... + a_kf_n-k)

Question 14

Which of the following is a disadvantage of multi-step methods?

Accepted Answer

They can be numerically unstable.

Answer

They are less complex to implement than single-step methods.

Answer

They are more accurate than single-step methods.

Answer

They require less storage than single-step methods.

Question 15

What is the primary purpose of the predictor-corrector method in multi-step methods?

Accepted Answer

To enhance accuracy by using an iterative approach.

Answer

To simplify the implementation of multi-step methods.

Answer

To extend the applicability of multi-step methods to higher-order equations.

Answer

To reduce the computational cost of multi-step methods.

Question 16

Which of the following methods is a multi-step method?

Accepted Answer

Adams-Bashforth

Answer

Runge-Kutta

Answer

Euler

Answer

Midpoint

Question 17

What is the key difference between explicit and implicit multi-step methods?

Accepted Answer

Explicit methods use previously calculated values, while implicit methods solve a system of equations for the current solution.

Answer

Explicit methods are more accurate than implicit methods.

Question 18

Which of the following is a common application of multi-step methods?

Accepted Answer

Solving stiff ordinary differential equations

Answer

Solving partial differential equations.

Answer

Integrating functions numerically.

Answer

Finding the roots of nonlinear equations.

Question 19

How is the global truncation error related to the local truncation errors in multi-step methods?

Accepted Answer

The global truncation error is the accumulation of the local truncation errors.

Answer

The global truncation error is the product of the local truncation errors.

Answer

The local truncation error is the same as the global truncation error.

Question 20

Which technique can be employed to estimate the error in a multi-step method?

Accepted Answer

Richardson extrapolation

Answer

Monte Carlo simulation

Answer

Finite element analysis

Answer

Quadrature

Question 21

What does the stability region of a multi-step method represent?

Accepted Answer

The region in the complex plane where the method is numerically stable.

Answer

The region in which the method is most accurate.

Answer

The range of values for which the method will converge.

Question 22

For the ordinary differential equation y' = y - 1, express the Adams-Bashforth method of order 2.

Accepted Answer

y_n+1 = y_n + h * (3 * f(t_n, y_n) - f(t_n-1, y_n-1)) / 2

Answer

y_n+1 = y_n + h * (f(t_n, y_n) - 3 * f(t_n-1, y_n-1)) / 2

Answer

y_n+1 = y_n + h * (f(t_n, y_n) + f(t_n-1, y_n-1)) / 2

Question 23

Which technique can be employed to estimate the error of a multi-step method?

Accepted Answer

Richardson extrapolation

Answer

Runge-Kutta methods

Answer

Euler's method

Answer

Taylor series expansion

Question 24

Which multi-step method combines prediction and correction?

Accepted Answer

Adams-Bashforth-Moulton

Answer

Trapezoidal rule

Answer

Euler's method

Answer

Runge-Kutta

Question 25

For stiff equations, which category of multi-step methods is generally more suitable?

Accepted Answer

Implicit methods (e.g., Adams-Moulton)

Answer

Any type of multi-step method

Answer

Explicit methods (e.g., Adams-Bashforth)

Answer

Single-step methods (e.g., Runge-Kutta)

Question 26

Which multi-step method is commonly applied in solving parabolic partial differential equations?

Accepted Answer

Crank-Nicolson method

Answer

Runge-Kutta method

Answer

Backward Euler method

Answer

Adams-Bashforth method

Question 27

What is the truncation error of the Adams-Bashforth method of order p?

Accepted Answer

O(h^(p+1))

Answer

O(h^(p+2))

Answer

O(h^p)

Question 28

Which of the following is a limitation of multi-step methods?

Accepted Answer

They may require additional starting values, increasing computational cost.

Answer

They are unconditionally stable.

Answer

They require step sizes to be smaller than single-step methods.

Question 29

Which of the following is a defining characteristic of a multi-step method?

Accepted Answer

Uses the values of the solution and its derivatives at multiple previous steps to approximate the solution at the next step.

Answer

Uses only the value of the solution at the current step to approximate the solution at the next step.

Answer

Is always more accurate than single-step methods.

Answer

Requires a fixed step size.

Question 30

In the context of multi-step methods, what is an implicit method?

Accepted Answer

A method that requires solving a nonlinear equation to determine the solution at the next step.

Answer

A method that has a stability region that includes the imaginary axis.

Answer

A method that is more accurate than an explicit method of the same order.

Answer

A method that uses only previous values of the solution to approximate the solution at the next step.

Question 31

Which of the following is a formula for the Adams-Bashforth method of order 3?

Accepted Answer

y_{n+3} = y_n + h * (23/12 * f(t_n, y_n) - 4/3 * f(t_n-h, y_n-h) + 5/12 * f(t_n-2h, y_n-2h))

Answer

y_{n+3} = y_n + h * (1/3 * f(t_n, y_n) + 2/3 * f(t_n-h, y_n-h))

Answer

y_{n+3} = y_n + h * (1/4 * f(t_n, y_n) + 3/4 * f(t_n-h, y_n-h))

Answer

y_{n+3} = y_n + h * (1/12 * f(t_n, y_n) + 1/2 * f(t_n-h, y_n-h) + 1/12 * f(t_n-2h, y_n-2h))

Question 32

In the context of multi-step methods, what is the purpose of a predictor-corrector method?

Accepted Answer

To improve the accuracy of the solution by using a lower-order method to predict the solution at the next step and then using a higher-order method to correct the prediction.

Answer

To increase the unconditional stability of the method.

Answer

To reduce the computational cost of the method.

Answer

To allow for the use of non-uniform step sizes.

Question 33

Which of the following is true about the Adams-Moulton method?

Accepted Answer

It is a two-step method.

Answer

It is a one-step method that uses only past values of the solution.

Answer

It is a multi-step method that uses only future values of the solution.

Question 34

What is the order of accuracy of the Adams-Bashforth method?

Accepted Answer

Two

Answer

One

Answer

Four

Answer

Three

Question 35

Which of the following is the formula for the Adams-Moulton method for solving first-order ordinary differential equations?

Accepted Answer

$$y_{n+1}=y_n+h(f_{n+1}+f_n)/2$$

Answer

$$y_{n+1}=y_n+hf_{n+1}$$

Answer

$$y_{n+1}=y_n+h(f_n+f_{n-1})/2$$

Answer

$$y_{n+1}=y_n-hf_{n+1}$$

Question 36

Which of the following is a multi-step method that is implicit?

Accepted Answer

Adams-Moulton method

Answer

Adams-Bashforth method

Answer

Euler method

Answer

Runge-Kutta method

Question 37

Which of the following is not a factor to consider when choosing a multi-step method?

Accepted Answer

The number of stages

Answer

The stability region

Answer

The order of accuracy

Answer

The computational cost

Question 38

Which of the following is not a multi-step method for solving ordinary differential equations?

Accepted Answer

Euler's method

Answer

Backward differentiation formula (BDF)

Answer

Adams-Moulton method

Answer

Adams-Bashforth method

Question 39

What is the order of accuracy of the Adams-Bashforth 2-step method?

Accepted Answer

2

Answer

1

Answer

4

Answer

3

Question 40

Which of the following is a disadvantage of Adams-Moulton methods compared to Adams-Bashforth methods?

Accepted Answer

They require more function evaluations

Answer

They are unstable for stiff equations

Answer

They are more sensitive to step size

Answer

They are less accurate

Question 41

What is the fundamental difference between an explicit and an implicit multi-step method?

Accepted Answer

An explicit method can be solved without solving a system of equations, while an implicit method requires solving a system of equations

Answer

An explicit method is more accurate than an implicit method

Answer

An explicit method is always stable while an implicit method is always unstable

Answer

An implicit method requires more computational cost than an explicit method

Question 42

Which of the following conditions must be satisfied for a multi-step method to be convergent?

Accepted Answer

The method must be stable and consistent

Answer

The method must be explicit

Answer

The method must be of high order

Answer

The method must be self-starting

Question 43

What is the primary advantage of using a multi-step method with a higher order of accuracy?

Accepted Answer

It requires fewer steps to achieve the same level of accuracy

Answer

It can solve a wider range of problems

Answer

It is more stable than a lower order method

Answer

It is less sensitive to step size