7) Numerical Solutions of PDEs

Question 1

When is a function said to be O(h^n)

Answer 1

Question 2

What is the Foward Difference Method

Answer 2

Question 3

Describe the proof of how the forward difference works

Answer 3

Question 4

When is a finite-difference method said
be nth order

Answer 4

If the remainder term is O(h^n)

Question 5

What is the Centred Differencing Method

Answer 5

Question 6

Describe the proof of the Centred Differencing Method

Answer 6

Question 7

What is the advantage of using centred differencing

Answer 7

It is higher order, but still only requires the same number of evaluations of u

Question 8

How do you construct a finite-difference approximation for the first derivative of a function using Taylor’s theorem

Answer 8

Question 9

Why are the coefficient conditions a+b+c+d=0,
a−c−2d=1, and a+c+4d=0 necessary in finite-difference formulas for first and second-order accuracy

Answer 9

First-Order Accuracy:
* Sum Zero (a+b+c+d=0): Ensures cancellation of constant terms, focusing the formula on the derivative approximation.
* Linear Coefficient (a−c−2d=1): Scales the first derivative term to 1, aligning the approximation with u′(x)
Second-Order Accuracy:
* Quadratic Coefficient Zero (a+c+4d=0): Removes the second derivative term from the error, reducing it to O(h^2 ) and increasing accuracy.

Question 10

How is the second derivative of a function approximated using the central finite-difference method, and what is its accuracy

Answer 10

Question 11

What is the Explicit Euler Method

Answer 11

Question 12

Describe the proof of the Explicit Euler Method

Answer 12

Question 13

What is the Implicit Euler Method

Answer 13

Question 14

What is the Theta Method

Answer 14

Question 15

How does the theta method generalise explicit and implicit Euler methods

Answer 15

Θ = 1 => Explicit Euler - (Error 0(∆t)
Θ = 0 => Implicit Euler - (Error 0(∆t)
Θ = 1/2 => Trapezoidal method - (Error 0(∆t^2)

Question 16

What is the Trapezoidal Method

Answer 16

Question 17

How is the Implicit Euler method applied to vector-valued ODEs

Answer 17

Question 18

What is a Boundary Value Problem (BVP)

Answer 18

A PDE with two boundary conditions (often either the values at the boundaries or the derivatives at the boundaries)

Question 19

How can finite-difference methods be used to approximate the solution of a boundary value problem (BVP)

Answer 19

Question 20

How is the vector b structured when adding forcing terms to finite-difference schemes for BVPs

Answer 20

Question 21

What is Global Error

Answer 21

The difference between the finite-difference approximation and the true solution, given at the gridpoints xj .
That is, ej = u(xj ) − uj , j = 0, 1, . . . , N

Question 22

When is a finite-difference method said to be convergent

Answer 22

If it is both stable and
consistent or

Question 23

What is the Truncation Error of a finite-difference relation

Answer 23

The local truncation error Tj at the grid point xj is the
remainder when uj is replaced by u(xj ) in the finite-difference relation for that point

Question 24

When is a finite-difference method said to be consistent

Answer 24

A finite-difference method is said to be k-th order consistent for k > 0 if the local truncation error satisfies:
Tj = O(h^k)

Question 25

What are the consistency orders of forward, backward and central finite-difference approximations

Answer 25

The forward and backward finite-difference approximations of the first spatial derivative are 1st order consistent
The central difference approximations of the first and second spatial derivatives are 2nd order consistent

Question 26

What is Stability

Answer 26

A numerical method is stable if it produces an approximation of the true solution, which exactly solves a “nearby” differential equation

Question 27

What is the Region of Absolute Stability

Answer 27

Question 28

What is the region of absolute stability for the explicit Euler method

Answer 28

∣1+λΔt∣< 1
In terms of the complex variable z=λΔt,
Thus the solution Xn converges to zero as n→∞ if and only if λΔt is within this defined circle

Question 29

What is the region for absolute stability for the implict Euler method

Answer 29

|1 − ∆tλ| > 1
In terms of the complex variable z=λΔt,
Thus the solution Xn converges to zero as n→∞ if and only if λΔt is within this defined circle

Question 30

What is the advantage of using the implicit Euler method over the the explicit method

Answer 30

This region for implicit Euller that converges is far bigger than the explicit method, and this means that we have a lot more flexibility in our choice of step size ∆t

Question 31

What is the region for absolute stability for the trapezoidal method

Answer 31

|1 +λ∆t/2|< |1 − λ∆t/2|

Question 32

What is the Stability Theorem for finite-difference Schemes for BVPs

Answer 32

Question 33

What do we know about a finite-difference method if the appoximation is both stable and kth order consistent

Answer 33

Question 34

How is the Method of Lines applied using finite differences to solve the heat equation in one spatial dimension

Answer 34

Question 35

What is the Crank-Nicolson method

Answer 35

Question 36

What is the convection-diffusion equation

Answer 36

Question 37

How can the convection-diffusion equation be solved

Answer 37

Use Central-difference approximation

Question 38

What is the mesh Peclet number

Answer 38

h|w|/ 2
The centred difference approximation is only stable when the mesh-Peclet number is less than 1

Question 39

What is the upwind finite-difference method

Answer 39

Question 40

What does it mean when the upwind finite-difference method is described as “unconditionally stable”

Answer 40

The term “unconditionally stable” in the context of the upwind finite-difference method implies that the numerical scheme remains stable regardless of the value of the grid spacing h, provided h>0