Maths Y2

Question 1

Laplace Equation

Answer 1

∇²φ = 0

Question 2

Poisson Equation

Answer 2

∇²φ = f

Question 3

Diffusion Equation

Answer 3

∇²φ - k ∂φ/∂t = 0

Question 4

Wave Equation

Answer 4

∇²φ - 1/C² ∂²φ/∂t² = 0

Question 5

Finite Difference for Laplace (Elliptic), u_xx(x,y)

Answer 5

u_xx(x,y) ≈ 1/h² ( u(x+h, y) - 2u(x, y) + u(x-h, y) )

(Do this for xx and yy then substitute into PDE)

Question 6

Jacobi Method

Answer 6

• Set up simultaneous equation matrix
• Rearrange to get expressions for each unknown, u
• Start with an initial guess for each unknown
• Iterate, using the previous iteration values of each u to find a new value for each u
• Repeat until within desired tolerance

Question 7

Gauss-Seidel Method
(or Liebmann’s Method)

Answer 7

• Set up simultaneous equation matrix
• Rearrange to get expressions for each unknown, u
• Start with an initial guess for each unknown
• Find new u values, using the previous iteration values of a u that has an index after the u being found currently, but u values from the same iteration if they have already been found in that iteration
• Repeat until within desired tolerance

Question 8

Iterative Methods accuracy criterion

Answer 8

( u_new) - u_old ) / u_new | < ε

Question 9

Finite difference (explicit) method for Diffusion Equation (Parabolic)

Answer 9

1/k ( u(x, t+k) - u(x, t) ) - 1/h² ( u(x+h, t) - 2u(x, t) + u(x-h, t) ) = 0
- k is timestep
- h is space-step
- r = k/h²
- 2k ≤ h²
u(x, t+k) = r( u(x+h, t) + u(x-h, t) ) + (1-2r) u(x, t)

Question 9

Crank Nicholson (Implicit) method for Parabolic Equation

Answer 9

1/k ( u(x, t+k) - u(x, t) ) = 1/2h² ( u(x+h, t) - 2u(x, t) + u(x-h, t) ) + 1/2h² ( u(x+h, t+k) - 2u(x, t+k) + u(x-h, t+k) )
- Same as Explicit method, but the u_xx part is averaged over 2 time steps
- Timestep k no longer restricted
- Can sometimes cancel out terms on either side if r = k/h² = 1

Question 10

Newton Raphson for coupled Equations

Answer 10

x_n+1 = x_n - J^-1_n f_n

• x = (x1, x2)^T
• J = Jacobian with derivatives of f
• f = Matrix of defining functions equal to 0

Question 11

How to solve coupled 2nd Order ODEs

Answer 11

• y1= y
• y2 = dy/dt
• Make a matrix transforming (y1, y2) into d/dt(y1, y2)
• Find Eigenvalues of that matrix
• Make an eigenvalue matrix for the equation w’_i = λ_i w_i
• Use standard forward Euler with w instead of y
• w converges if |1+λh| < |1|
• if w converges, then y converges

Question 12

Binomial Coefficient Function nCr

Answer 12

nCr = n! / ( r! (n-r)! )

Question 13

Euler Cromer Method

Answer 13

For coupled ODEs:
Rearrange system to make one variable solve explicitly (FWD Euler)
And one variable solved implicitly (BWD Euler, use gradient for n+1)
e.g. y(n+1) = y(n) + hy’(n)
or y(n+1) = y(n) + hy’(n+1)
To solve you can sub the explicit into the implicit and make a matrix with that and the regular explicit
(A semi-implicit method)

Question 14

Lagrange Interpolation

Answer 14

Making a polynomial that has the exact correct value for each node specifically, by making all the other terms 0
(x - x1)/(x0 - x1) = 0 for x = x1

Question 15

Newton’s divided difference interpolation

Answer 15

Just use the formula book, it’s got the differencey things
Remember for backwards, n is negative, so values are y₀, y_-1, y_-2 etc. And y₀ is the one that has all the higher order divided differences

Question 16

Cubic spline interpolation

Answer 16

Use the formula book notes for the equations, just find all the missing variables

Question 17

Bilinear interpolation

Answer 17

The one with the 4 points in a rectangle, and a simultaneous matrix for all of them
Can do without matrix if you solve generally for two vertical lines, then interpolate the vertical lines

Question 18

Barycentric Interpolation

Answer 18

Just use the formula book for the equations for finding λ
Can use matrix form A λ = r
- Where r is the interpolated point cartesian coordinates
- A is a matrix where each column is one of the triangular points
- λ is a vector of the 3 barycentric coordinate values

Question 19

Simpson’s Rule

Answer 19

Used for numerical integration
Requires an even number of subintervals, odd number of nodes

Question 20

Gauss Quadrature

Answer 20

Used for numerical integration
Use a substitution, sub in expression in t from formula book
Calculate f(t) where f is the original function, but with t subbed in
Compute the nodes & weightings and sum them up
A and B are the integration limits

Question 21

Forward Euler Method

Answer 21

Solve ODEs that are initial value problems
simply y(n+1) = y(n) + h y’(n)
f(y,t) often used to mean the derivative, y’

Question 22

Heun’s Method

Answer 22

Improved forward Euler used for ODEs
- Predicts next y, y(n+1) using FWD euler
- Finds gradient at y(n+1)*
- Averages with gradient at y(n)
- Uses this average as the gradient used in FWD euler to get y(n+1)

Question 23

Runge-Kutta Methods

Answer 23

Like Heun but with as many steps as you want, avergaing many gradients basically

Question 24

Multistep Methods

Answer 24

Use multiple initial values of y to get going

Question 25

4th Order Adam's Bashford

Answer 25

A multistep method for solving ODEs Derived from averaging multiple backwards divided differences

Question 26

Adam's Moulton Method

Answer 26

Same as Adam's Bashford but Implicit (using future y values) Uses forwards divided difference instead of backwards uses f(n+1), f(n+2) etc values

Question 27

Finite difference (explicit) for wave equation

Answer 27

set up u_xx = 1/h²(...... etc also set up u_tt = 1/k²(.... etc Replace both into normal PDE uxx = utt r = k²/r² r <= 1 is stable Known initial velocity allows for t = 0-k to be found using finite central difference between 0+k and 0-k

Question 28

Numerical Method is consistent when

Answer 28

lim(h→0) (ε_local/h) = 0