3) Numerical Solution of ODEs

Question 1

What is a Lipschitz condition for a function

Answer 1

Question 2

When can the Lipschitz constant L be taken as the partial derviate with respect to y

Answer 2

Question 3

What does Picard’s theorem state about the existence and uniqueness of solutions to differential equations

Answer 3

Question 4

What is Euler’s method for solving differential equations

Answer 4

Question 5

What is a one-step method and a multistep method in numerical integration

Answer 5

One-step Method - the step from (xn,yn) to (xn+1 ,yn+1) is computed using only the current values (xn ,yn), with no use of previously computed information (xk ,yk) for k < n
Multistep Method - use previously computed values

Question 6

What is the general form of a one-step method

Answer 6

Question 7

What is the Backward Euler method

Answer 7

Question 8

How does the Backward Euler method fit into the definition of a one-step method

Answer 8

Question 9

What is the Trapezium method for solving differential equations

Answer 9

Question 10

What is the local truncation error for a one-step method, and what does it mean for the method to be of order p

Answer 10

Question 11

How do we prove that Euler’s method has order p=1

Answer 11

Question 12

How do we show that the trapezium rule has order p=2

Answer 12

Question 13

What is the order of accuracy and the local truncation error for Euler’s method, the backward Euler method, and the trapezium method

Answer 13

Question 14

What does it mean for a method to be consistent

Answer 14

A method is consistent if it has order p≥1. This means that the local truncation error goes to zero as the step size h→0

Question 15

What is global error

Answer 15

The quantity en = y(xn) - yn

Question 16

What does it mean for a method to be convergent

Answer 16

Question 17

What is the global error for Euler’s method at any grid point xn

Answer 17

Question 18

How do we derive the global error bound for Euler’s method

Answer 18

Question 19

What is Heun’s method (also known as the improved Euler method)

Answer 19

Question 20

What is the general form of a 2-stage Runge–Kutta method and the general m-stage Runge–Kutta method

Answer 20

Question 21

How do you do the Taylor expansion of a multivariable function

Answer 21

f(xn + Δx,yn +Δy) =f(x ,yn)+Δx⋅fx (xn,yn)+Δy⋅fy(xn,yn)

Question 22

What are the conditions for the 2-stage Runge–Kutta method to be of order 2

Answer 22

Question 23

What are embedded Runge–Kutta methods and how do they help with error estimation

Answer 23

Question 24

What is the general form of Linear Multistep (LM) methods

Answer 24

Question 25

What are the Adams methods

Answer 25

Question 26

What is the 1-step Adams–Bashforth method

Answer 26

Question 27

What is the 2-step Adams–Bashforth method

Answer 27

Question 28

What are the formulas for the 3-step and 4-step Adams–Bashforth methods

Answer 28

Question 29

How are Adams–Moulton methods different to Adams-Bashford methods and how are they built

Answer 29

It approximates the next value yn+1 using past values of f(x,y) and also includes f(xn+1,y n+1), making it more accurate but requiring solving an equation each step. It’s often used as a corrector after a predictor like Adams–Bashforth

Question 30

What is the 1-step Adams–Moulton method

Answer 30

Question 31

What is the region of absolute stability

Answer 31

The region of absolute stability is the set of hλ ∈C C for which yn → 0 as n → ∞

Question 32

What is an A-stable method

Answer 32

If the region of absolute stability includes the entire left halfplane, then the method is said to be A-stable

Question 33

What is the region of absolute stability for Euler’s method, backward Euler method, trapezium method, and 2-stage second order Runge-Kutta methods

Answer 33

Question 34

Why can't any m-stage Runge–Kutta method be A-stable

Answer 34

Question 35

How do we solve the nonlinear equation in implicit methods, and what ensures convergence

Answer 35

Question 36

When does fixed point iteration converge

Answer 36

Question 37

How does Newton’s method work for solving F(y)=0 (use backward Euler in this example)

Answer 37

Question 38

What is a predictor-corrector method and why is it used

Answer 38

A predictor-corrector method uses: * An explicit method (predictor) to make a quick guess for yn+1, and * An implicit method (corrector) to refine the guess * This helps improve accuracy and speed up convergence

Question 39

How does the PECE predictor-corrector method work for Euler + Trapezium

Answer 39

Question 40

How does the PECE predictor-corrector method work for Adams–Bashforth + Adams–Moulton

Answer 40

Question 41

How do we handle higher-order ODEs and vector systems numerically

Answer 41