Overview
1
Recap
2
Partial differential equations
Classification
Finite difference methods
Boundary conditions
3
Summary
Quiz next week!
Recap
Many boundary-value odes can be discretised, made into matrix
equations
Three general methods for solving matrix equation Ax = b:
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Direct elimination (Gauss–Jordan etc)
Iteration (Gauss–Seidel etc) for diagonally dominant matrices
Conjugate gradient type for sparse matrices
Sparse matrices appear in many physical problems
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Huge savings in storage and computation
Many numerical methods are tailor-made for sparse matrices
Partial differential equations
A very large number of physical problems are formulated as pdes:
Schrödinger equation
i~
i
∂Ψ h ~2 2
→
= −
∇ + V (−
x) Ψ
∂t
2m
Poisson equation
→
∇2 u = ρ(−
x)
Navier–Stokes equation
ρ(
→
∂−
u
1
→
→
→
→
+−
u∇·−
u ) = −∇p + η∇2 −
u + ( η + ζ)∇(∇ · −
u)
∂t
3
Korteweg–de Vries equation
∂φ ∂ 3 φ
∂φ
+ 3 + 6φ
=0
∂t
∂x
∂x
and many more!!
Complications, simplifications
ODE: One variable, one general theory, general methods
PDE: Several mathematically very different cases!
We have 3 (10?) space dimensions, but only one time dimension
Equations must be 0th, 1st or 2nd order in time:
ṗ = −
and p = p(q, q̇, t)
∂H
∂q
Classification
Traditional classification
Second-order pde, 2 variables:
a
∂2u
∂2u
∂2u
∂u
∂u
+
2b
+
c
+d
+e
+ fu + g = 0
∂x 2
∂x∂y
∂y 2
∂x
∂y
elliptic
parabolic
hyperbolic
b 2 − ac > 0 Poisson
b 2 − ac = 0 Diffusion
b 2 − ac < 0 Wave
Not very useful from computational view
Classification
Computational classification
Time evolution (initial value problem)
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Mostly hyperbolic, parabolic
Most important issue: stability
Static solution (boundary value problem)
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Mostly elliptic
Most important issue: efficiency
A lot of pdes are a mix of parabolic, hyperbolic, elliptic!
Methods
Finite differences
Finite elements
Monte Carlo
Spectral methods (later!)
Variational
...
Finite differences
1
Replace x and y with a discrete grid (i, j):
xi = x0 + iδx ,
yj = y0 + jδy
Function is given by values on lattice points: u(x, y ) → uij
2
3
Replace derivatives by finite differences — but how?
Result can be written as matrix equation Au = b
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4
u, b are N = nx × ny vectors
A is a sparse N × N matrix
Use direct/iteration/Fourier methods to solve for u
In isotropic systems (with static solutions) we usually choose δx = δy = a
but for time evolution we need δt δx
Poisson equation
∂2Φ ∂2Φ
+
= ρ(x, y )
∂x 2
∂y 2
Discrete derivative:
=⇒
∂2Φ
∂x 2
→
Φi+1,j − 2Φi,j + Φi−1,j
a2
Φi+1,j + Φi−1,j + Φi,j−1 + Φi,j+1 − 4Φi,j = a2 ρij ≡ ρ̂ij
To write as matrix equation,
relabel coordinates: k = i + Nx j
d
d
w
d
d
We get AΦ = ρ̂
Amn = δm,n+1 + δm,n−1 + δm,n+Nx
+ δm,n−Nx − 4δmn
A is sparse: most Amn = 0!
Boundary conditions
Three important classes of boundary conditions:
Dirichlet Φ(x) = b(x) on boundary
Neumann ∂n Φ(x) = b 0 (x) on boundary
∂n Φ is normal derivative, orthogonal to boundary
Periodic Φ(x + L) = Φ(x)
This only works for regular (rectangular) boundaries!
Used mostly when we are interested in bulk behaviour
→ take large volume limit
Topology changes: line → circle, rectangle → torus, . . .
Implementation
Dirichlet
If one of the neighbours is on the boundary, it gets replaced by the
boundary value bj in the finite difference:
∇2 Φ1,j
→
b0j + Φ2,j + Φ1,j−1 + Φ1,j+1 − 4Φ1,j
Recipe:
Modify finite diff on edges to include only ‘available’ directions
Subtract (sum of) boundary values from source term
Implementation
Neumann
Use the forward/backward derivative on the boundary:
0
b0j
= (∂Φ)0j = Φ1j − Φ0j
0 in finite diff operator
So we replace Φ0j → Φ1j − b0j
Recipe:
Modify finite diff operator on edges to replace ‘unavailable’ direction
‘Φ0 ’ by Φk
Add (sum of) boundary values to source term
Periodic
If we are at the edge, the nearest neighbour is at opposite edge!
So Φ0j → ΦLj ; ΦL+1,j → Φ1j etc in all finite differences
Irregular boundaries
We start by mapping our boundary onto the regular grid Number interior
grid points and boundary points Find distances from interior points to
boundary Taylor expand differential operators near boundary
t
tt
b5
........................................................
....................
........
13
14 .................
.....
....
.
.
.
.
....
.
.
.
.....
.
...
.
.
.
.
.
.
...
b....3.........
...
.
.
9
10
11
12 ....
b2 ...... 8
.
..
.
.
.
.
...
.
..
.
.
...
...
..
..
.
.
.
.
.
.
.
.
..
..
...
b1 ....
4
5
6
7
...
..
....
..
...
.
..
.
.
...
....
...
.....
......
...
.......
....
.......
.
.
.....
.
.
.
.
.
...
2
3
....... 1
..........
........
...........
...........
..............
..................
.......................................................
t ..........
θ
b..4......................
tt
t
t
tt
t
tt
t
t
tt
Repeat for all boundary points!
Taylor expansion around Φ9 :
1
b4 ≈ Φ9 + θ∂y Φ9 + θ2 ∂y2 Φ9
2
1
Φ5 ≈ Φ9 − ∂y Φ9 + ∂y2 Φ9
2
θΦ
+
b
−
(1
+ θ)Φ9
5
4
∂y2 Φ9 ≈
θ(1 + θ)
Some MatLab tricks
reshape(X,M,N)
reshape(X,M,N,P)
rearranges X into a M×N matrix
rearranges X into a M×N×P matrix
etc.
[X,Y] = meshgrid(x,y)
numgrid(region,N)
delsq(G)
speye(N)
creates
creates
creates
creates
del2(U)
evaluates the laplacian of U
2d arrays X,Y from vectors x,y
a numbered grid for various regions
the 2-dim laplacian on such a grid
the sparse N×N unit matrix
Summary
Classification of pdes:
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time evolution, initial value (mostly parabolic, hyperbolic)
static solution, boundary value (mostly elliptic)
Finite differences transform pdes into matrix equations
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static solution problems may be solved directly
Boundary conditions
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Dirichlet: subtract boundary terms from source term
Neumann: modify finite diff on boundary, add boundary term to source
Periodic: wrap finite differences around boundary