1
Partial differential equations
• Numerical Solution of Partial Differential Equations, K.W. Morton and D.F. Mayers (Cambridge Univ. Press, 1995)
• Numerical Solution of Partial Differential Equations in Science and Engineering, L. Lapidus and G.F. Pinder (Wiley, 1999)
• Finite Difference Schemes and Partial Differential Equations, J.C. Strikwerda (Wadsworth, Belmont, 1989)
= PDE
2
Examples for PDEs
0
00 n
( ), ( )
, ( )
x
x
field depends on
Poisson equation:
Laplace equation:
examples for
scalar
boundary . value
problems
(elliptic eqs.)
Dirichlet boundary condition
von Neuman boundary condition
3
Examples for PDEs
1 0( ( )) ( ) ( )u x u x
example: vectorial boundary value problem
( )u x
is a vector field defined on space
Lamé equation of elasticity(elliptic eq.)
4
Examples for PDEs
)(),(
)(~
),( ,
0
002
2
2
t
tt
xtxct
wave equation
diffusion equation
initial boundary problem
),( tx
5
Examples for PDEs
0 0 0 0
0 0
10( ) ,
( , ) ( ) , ( , ) ( )
( , ) v ( ) , ( , ) p ( )
vv v p v v
t
v x t V x p x t P x
v t t p t t
( , )v x t
vector field in space and time
Navier – Stokes eq. for fluid motion
x x xx y z
y y yx y z
z z zx y z
v v vv v v
x y z
v v vv v v
x y z
v v vv v v
x y z
6
Discretization of space
i,j = (xi ,yj)i,j
xi+1=xi+Δx
yj+1=yj+Δy
Finite
Difference
Method
7
Discretization of derivatives
1
1
21 1
2
small ,
( ) ( )( )
( ) ( ) ( )
( ) ( ) ( )
n
n n
n n
n n
x x n x
x xO x
x xx x
O xx
x xO x
x
first
derivative
in 1d
8
Discretization of derivatives
221 1
2 2
2( ) ( ) ( )( )n n nx x x
O xx x
second derivative in one dimension
242 1 1 2
2 2
16 30 16
12
( ) ( ) ( ) ( ) ( )( )
n n n n nx x x x x
O xx x
or better
9
Discretization of derivatives
)()(6
)(2
)()()( 43
33
3
2
22
2
xOxx
xk
xx
xk
xx
xkxx nnnnkn
1( )
i l
k n ki ik l
a xx x
)( 12
)()( 16)( 30)( 16)( 42
21122
2
xOx
xxxxx
xnnnnn
322 1 1 2
3 3
2 2( ) ( ) ( ) ( )( )n n n xx x x x
O xx x
insert in Taylor expansion:
i = 2
third derivative
10
Derivatives in higher dimension
21 1
1 1 4
( , ) ( , )
( , ) ( , ) ( , )n n n n
n n n n n n
x x y x y
x y x y x y
21
1 1 1
1 1 6
( , , )
( , , ) ( , , ) ( , , )
( , , ) ( , , ) ( , , )
n n n
n n n n n n n n n
n n n n n n n n n
x x y z
x y z x y z x y z
x y z x y z x y z
2 d
3 d
Be Δx = Δy = Δz .
11
Poisson equation
)()( xx
)( nn x
)( 2 211 nnnn xx
bediscretization of the Poisson equation:
discretize one-dimensional space by xn , n = 1,…,N
0 0 1and Nc c Dirichlet boundary conditions:
System of N-1 coupled linear equations
12
Poisson equation in 1d
1 0
2
3
14
2 1 0 0
1 2 1 0 0
0 1 2 1 0
0 0 1 2
c
c
example: chain of N = 5 with ρ = 0
and Dirichlet boundary conditions
15
Poisson equation in 2d
jijijijijiji x ,2
,1,1,,1,1 4
21 1 2 2 4 k k k L k L k kx
two-dimensional discretized equation on grid L L:(Δx = Δy)
replace indices i and j by k = i + ( j -1) (L-2)
System of N = (L-2) 2 coupled linear
equations:bA
16
Laplace equation in 2d
4 1 0 1 0 0 0 0 0
1 4 1 0 1 0 0 0 0
0 1 4 0 0 1 0 0 0
1 0 0 4 1 0 1 0 0
0 1 0 1 4 1 0 1 0
0 0 1 0 1 4 0 0 1
0 0 0 1 0 0 4 1 0
0 0 0 0 1 0 1 4 1
0 0 0 0 0 1 0 1
1
2
3
4
5
6
7
8
9
0
4
21
21
01
212
Example 5 5 lattice with ρ = 0 and m = 0 for all m Γ,
i.e. Dirichlet boundary condition with fixed 0 on Γ.
(L-2)2 (L-2)2 matrix
17
Exact solution
bA
1*
111 1 1
1 1
. .
. . . . .
. . . . .
. .
N N
N NN N N
ba a
a a b
'
'
, ,
ikik
kk
jl jl jk kl
i i ik k
aq
a
a a q a j l k
b b q b
1
1 1
1
*
* *
NN
NN
N
i i jj jj iii
b
a
b aa
solution
Gauss elimination procedure matrix A triangular
for k = 1,..., N once matrix
is triangular
O (N 3) ~ O (L3d)
bA
18
Poisson equation in 2d
Independently of the size of the system
each row or column has only
maximally five non-zero matrix elements
sparse matrix
Use sparse matrix solvers !
Invert with LU decomposition
19
Sparse matrices
example:
Hanwell Subroutine LibraryIain Duff
Store non-zero elements in a
vector and also their coordinates
i and j in vectors.
Yale Sparse Matrix Format
For more details see:www.cise.ufl.edu/research/sparse/codes
20
Sparse matrix solvers
21
Quadtrees
Tree data structure where
each node has up to four
children corresponding
to the four quadrants.
That means that each
node can contain several
pointers indexed by two binary variables
representing coordinates i and j.
22
Computational considerations
Computational effort for Gauss elimination N 3.
For a lattice 100 100 = 104 one needs 2 days.
Abandon exact solution and use approximation.
But for that A must be well-conditioned:
example for ill-conditioned situation:
23
Jacobi relaxation method
1 1 1 1
11
4
( ) ( ) ( ) ( ) ( )
( , ) ( , )
ij i j i j ij ij ijt t t t t b
x y b x y
1 1 1 1
1
4 * * * * *ij i j i j ij ij ijb
11
( )
( ) ( ) ( )
A b A D O U
D b O U
t D b O U t
fixed point is the exact solution:
decompose:
example 2d Poisson equation: start with any ij(0)
general:
24
Error of Jacobi relaxation
11
( ) ( )'( )
( )
t tt
t
Exact solution is only reached for t→∞.
Define required precision ε
and stop when :
1 1 1
1 1 1
1 1
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
exact solution approximate solution
t
t A b t A b D b O U t
D O U A b t D O U t
real error:
AA-1
25
Error of Jacobi relaxation
11( ) ( ) ( )t t D O U
tct
*)(
with
be λ the largest eigenvalue of Λ
1
1
1
1
( ) ( )
( ) ( )
t t
t t
t t
t t
0 < |λ| < 1
for large t :
26
Error of Jacobi relaxation
1 1 '( ) ( ) ( )t t
2
11
1 1 1
( ) ( )'( )( )
( ) ( ) ( ) ( ) ( )
t ttt
t t t t t
111 1
( ) ( ) ( )'( )
( ) ( ) ( )
t t
tt t c c
tt t t
* ( )( )
( ) ( )t
t ct
t t
real error:
27
Gauss-Seidel relaxation
11
( )
( ) ( ) ( )
A b A D O U
D O b U
t D O b U t
1
1 1
11 1( ) ( ) ( )
N i
i ij j ij j jj i jii
t a t a t ba
and
fixed point is the exact solution
28
Error in Gauss-Seidel
1 1
1 1 1
1
( )
( ) ( ) ( )
( ) ( ) ( ) ( )
exact solutionapproximate solution
t
t A b D O b U t
D O U A b t D O U t
11( ) ( ) ( )t t D O U
with
1
1
( ) ( )( )
( ) ( )
t tt
t
λ largest EV of Λ
29
Partial differential equations (PDF)
bA
Jacobi relaxation
Gauss-Seidel relaxation
30
Gauss-Seidel relaxation
11
( )
( ) ( ) ( )
A b A D O U
D O b U
t D O b U t
1
1 1
11 1( ) ( ) ( )
N i
i ij j ij j jj i jii
t a t a t ba
and
fixed point is the exact solution
31
Overrelaxation
Successive overrelaxation = SOR
11 1( ) ( ) ( ) ( )t D O b D U t
Fixed point is the exact solution.
ω is the overrelaxation parameter.
1 ≤ ω < 2
ω = 1 Gauss-Seidel relaxation Applet
32
Non-linear problem
1 1 1 1 0( ) ( ) ( ) ( )i j ij ij i j ij ij ij ijf U U f U U f U U f U U
Consider a network
of resistors with a
non-linear I-U relation f.
Then Kirchhoff‘s law
takes the form:
Solve with relaxation:
1 1
1 1
1 1
1 1 0
( ( ) ( )) ( ( ) ( ))
( ( ) ( )) ( ( ) ( ))
i j ij ij i j
ij ij ij ij
f U t U t f U t U t
f U t U t f U t U t
33
Gradient methods
1 AbbAAAr
Be matrix A positive and symmetric.
The residuum
is a measure for the error.
Minimize the functional:
otherwise 0
if 0 *1
rAr t
error
35
Gradient methods
exact solution
Φ(1)
Φ(2)
36
Gradient methods1 1 2( ) ( ) t t tb A A b A b A b A b
i iidα
1 2 2 2 t t t t t ti i i i i i
2i i ii ib A b A d A d A dα αb bα d
2 0
t
t i ii i i t
i ii i
i
d rd A d r
d A dα α
α
Be i the i th approximation.
Minimize along lines:
minimization condition with respect to i:
38
Method of steepest descent
1 1
2
1
1
1
Start with and choose
iterate: ,
i i
ii i i t
i i
i i i i
i i i i
d r
r b A
ru Ar
r u
r
r r u
each step N 2, but when matrix A sparse N
39
Gradient methods
41
Conjugate gradient
0 if ti jd Ad i j
1 , , i ii i i i i i it
i i
r dr b A α d
d Ad
1
11
i
i j jj
r b A d
Hestenes and Stiefel (1957)
Choose di conjugate
to each other:
as before:
→
42
Conjugate gradient
1
1 11
, tij i
i i jtj j j
d Ard r d r d
d Ad
0 if ti jr Ad i j
one can also show:
Construct conjugate basis using
an orthogonalization procedure:(Gram – Schmidt)
43
Conjugate gradient
1 1 1 1 , r b A d r
1
1
1 1 1 1 1
, ,
,
ti i i i i i i i i
i i i i i i i
c d Ad cd r d
r b A d r cr Ad d
ti ir r
1. initialize:
2. iterate:
3. stop when:Applet
44
Conjugate gradient
If matrix not symmetric then use
biconjugate gradient method.
and tr b A r b A
Consider two residuals:
This method does not always converge
and can be unstable.
45
Biconjugate gradient
1 1 1 1
1 1 1 1
,
, t
r b A d r
r b A d r
1 1
1 1
11
with and
, ,
, ,
t ti i i i i i i i i i
ti i i i i i i i i i i
t ti i i i
r r A d r r A d c r r
d r d d r d c r r
c d Ad c d d
ti ir r
1. initialize:
2. iterate:
3. stop when: 1 n
n i ii
α d
46
Preconditioning
1 1such that P AP
1 1 A bP P
ijiiij
iiijiiij A
PjiA
AP
1
otherwise 0
if 1
Choose a preconditioning matrix
and solve equation:
example: Jacobi preconditioner:
47
Preconditioning
1
1
2
D DP U D O
example: SOR preconditioner:
Preconditioned Conjugate Gradient
48
Multigrid procedureAchi Brandt (1970)
Consider coarser
lattices on which
the long-wave
errors are
damped out. h = 2
49
Multigrid procedure
1 , n n n nr b A A r
Strategy: solve the equation for
the error on the coarser lattice.
1. Determine residuum r on the original lattice.
W.L. Briggs, A Multigrid Tutorial
(Soc. For Ind. & Appl. Math, 1991)
Two-level procedure:
50
Multigrid procedure
1
ˆ ˆ ˆn nA r
1 1 ˆ n n
P
3. Then obtain the error on the
coarser lattice solving equation:
4. Then get the error on the
original lattice through an
extension operator P :
2. Define the residuum on
the coarser lattice through
a restriction operator R :
ˆ n nr r
R
51
Multigrid procedure
1 1n n n
5. Get new approximate solution through:
In an m-level procedure one solves the
equation only on the last (coarsest) level.
On each level one can also smoothen the error
using several Gauss-Seidel relaxation steps.
52
Multigrid procedure
2 2
2 1 2 1
2 2 1 1
2 1 2 1 1 1 1 1
1
21
21
4
, ,
, , ,
, , ,
, , , , ,
ˆ
ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ ˆ
i j i j
i j i j i j
i j i j i j
i j i j i j i j i j
r r
r r r
rr r r
r r r r r
P
Example for extension operator on square lattice:
bilinear interpolation
53
Multigrid procedure
1 1 1 1
1 1 1 1 1 1 1 1
1 1
4 81
16
, , , , , ,
, , , ,
ˆ
i j i j i j i j i j i j
i j i j i j i j
r r r r r rr
r r r r
R
Corresponding restriction operator:
They are adjunct to each other, i.e.
2 ̂̂, ,
ˆ ˆ ˆ ˆ ˆ ˆ ( , ) ( , ) ( , ) ( , )x y x y
v x y u x y h v x y u x yP R
54
Multigrid procedure
One can also vary the protocol
V-cycles, W-cycles, …
55
Solving PDEs
A b discretize system of
coupled linear equations
● Finite difference methods:
Field is discretized on sites: i .
● Finite element methods = FEM:
Field is patched together from a discrete
set of continuous functions.
56
The Fathers of FEM
J. Argyris R.W. Clough O.C. Zienkiewicz
57
Finite Elements at ETH
• Gerald Kress: Strukturanalyse mit FEM
• Christoph Schwab: Numerik der Dgln.
• Peter Arbenz: Introduction to FEM
• Pavel Hora: Grundlagen der nichtlinearen FEM
• Andrei Gusev: FEM in Solids and Structures
• Falk Wittel: Eine kurze Einführung in FEM
• Eleni Chatzi: Method of Finite Elements
58
Literature for FEM
● O.C. Zienkiewicz: „The Finite Element
Method“ (3 Volumes), 6th edition
(Butterworth-Heinemann, 2005)
● K.J. Bathe: „Finite Element Procedures“
(Prentice Hall, 1996)
● H.R. Schwarz: „Finite Element Methods“
(Academic Press, 1988)
59
Finite Elements
Strukturmechanik/Anwendung:[6] J. Altenbach und U. Fischer: Finite-Elemente Praxis,
Fachbuchverlag Leipzig (1991)
[7] P. Fröhlich: FEM-Anwendungspraxis. Einstieg in die
Finite Elemente Analyse, Vieweg Verlag (2005)
[8] B. Klein: FEM, Vieweg-Verlag 6. Aufl. (2005)
[9] K. Knothe and H. Wells: Finite Elemente, Springer-Verlag (1991)
[10] F.U. Mathiak: Die Methode der finiten Elemente (FEM) –
Einführung und Grundlagen (2002).
[11] G. Müller und I. Rehfeld: FEM für Praktiker, Expert-Verlag (1992)
[12] M. Link: Finite Elemente in der Statik und Dynamik,
Teubner-Verlag 3. Aufl. (2002)
[13] H. Tottenham und C. Brebbia: Finite Element Techniques in
Structural Mechanics, Southhamptom.
[14] R. Steinbuch: Finite Elemente - Ein Einstieg, Springer-Verlag (1998)
60
Properties of FEM
• Irregular geometries
• Strongly inhomogeneous fields
• Moving boundaries
• Non-linear equations
Advantage of finite elements over finite differences
adaptive meshing, e.g. triangulation
Clough (1960)
61
Adaptive meshing in 2d
triangulations with different resolution
62
Adaptive meshing in 3d
triangulation of a wheel-rim
63
One dimensional example
2
24 0with 0
dx x L
dx
Poisson equation:
1 1
N
i i N i ii i
x a u x x a u x
Expand in terms of localized basis functions ui:
64
One dimensional example
2
20
4 0 1
, , ...,L
j
dx x w x dx j N
dx
baA
2
20 0
4 1 , ,...,
L L
j j
dx w x dx x w x dx j N
dx
Define weight functions wj (x) and get ai from:
wj (x) = uj (x) is called the Galerkin method.
system of linear equations
2
210 0
4 1
, ,...,L LN
ii j j
i
ua x w x dx x w x dx j N
x
2
21 0 0
4 1
, ,...,L LN
ii j j
i
ua x w x dx x w x dx j N
x
65
One dimensional example
0 0
''( ) ( ) '( ) '( ) L L
ij i j i jA u x w x dx u x w x dx
0
4 ( ) ( ) L
j jb x w x dx
baA
with
and
66
One dimensional example
1 1
1 1
( ) / for ,
( ) / for ,
0 otherwise
i i i
i i i i
x x x x x x
u x x x x x x x
Example for basis functions ui(x) are hat functions
centered around xi:
Δx ≡ xi – xi-1
=„element“
xi
ui(x)
xi-1 xi+1
67
One dimensional example
0 10 , L
Boundary conditions are automatically fulfilled
because basis functions were zero at both ends.
0 11
1
N
N i ii
x L x x a u xL
If
then use following decomposition:
68
Non-linear PDEs
2
24
dx x x
dx
2
20
4 0 L
k
dx x x w x dx
dx
,ijk i j k
i j
A a a b 0
'' L
ijk i j kA u x u x w x dx
1d example :
Then solve:
with
i.e. the coupled non-linear system of equations:
69
Picard iteration
21
0 24( ) ( ) ( )
dx x x
dx
21
24( ) ( ) ( )n
n
dx x x
dx
Start with a guess Φ0 .
Solve linear equation for Φ1 :
Then iterate:
Émile Picard
70
Finite Elements
1
, ,n
i ii
x y N x y
Decompose in basis functions Ni
0 ( , )x y a b
71
Variational Approach
dsdxdyaE
G
2 b
2
1
2
1 222
G
dsdxdyaE b
G Gds
ndxdydxdy
Minimize the functional: Argyris (1954)
first Green‘s theorem:
= 0
72
Variational Approach
ba
2 21 1
2 2 b
G
E a dxdy
1
2 E A b
0 0 E
A b
a = 0 Poisson equation
b = 0 Helmholtz equationFirst term
of total
energy
can be brought
into the form:
Minimizing
then gives:
2 21 1
2 2
b jG
elements j
E a dxdy
74
Function on Element
1 2 3 ( )r a a x a y
2 21 2 3 4 5 6 ( )r a a x a y a x a xy a y
Higher dimensions
In 2d define function over
one element = triangle of the triangulation
e.g. linearly:
or by a paraboloid:
75
Linear case
76
Standard Form
Transform any element j
into the standard form.
1
1
T
η
ξ
77
Coordinate transformation
yyxxyx
,,
D
xx
yD
yy
xD
xx
yD
yy
x12121313
≡ ≡
78
Coordinate transformation
... ...det jG T
dxdy J d d
yy
xx
J
2 1 3 1 3 1 2 1
detx y x y
J
x x y y x x y y D
Jacobi matrix
79
Coordinate transformation
2 2 2 21 2 32
j Tx yGdxdy c c c d d
Inserting gives for each element
where the
coefficients
are only
calculated
once for
each element.
2 2 2 21 2 32
j T TxG Ty dxdy c d d c d d c d d
80
1 2 3 ( )r a a x a y
2 21 2 3 4 5 6 ( )r a a x a y a x a xy a y
In 2d define function over
one element = triangle of the triangulation
e.g. linearly:
or by a paraboloid:
Basis functions
81
Shape of basis functions
82
Basis functions
6
1
, , ,i ii
N N
1 2
3 4
5 6
1 1 2 2 2 1
2 1 4 1
4 4 1
,
,
,
N N
N N
N N
1 6 1 6, ..., , , ...,N N N
Decompose on standard element in basis functions Ni
83
Shape functions on square lattice
2 2 2 2 2 21 2 3 4 5 6 7 8 9 ( )r c c x c y c x c xy c y c xy c x y c x y
85
Energy Integrals
1
22
1 ,
T T
t t t t
T T
S
I d d N d d
N N d d N N d d
2 2
23 3
t
T
t
T
I d d S
I d d S
Calculate the energy integrals on standard element
defining matrices
S1, S2 and S3 on
standard triangle.
and analogously
6 x 6
86
Rigidity Matrix
2 21 2 32
2
jG
t
T
dxdy
c c c d d S
1 1 2 2 3 32S c S c S c S
defines the rigidity matrix S for any element:
87
Mass Matrix
22
( , )
jj jG T
t t tj j j j j jT
a dxdy a N D d d
a N N D d d M
2 2
j
tj j j jG
elements j elements j
E a dxdy S M
Analogously one defines the mass matrix M:
with and tj j j
jE A A S M
Mj
assembly
88
Assembly of the Matrix
The elements must
be joined such that
the field is continuous.
This is done by identifying
the values of the coefficients
at each vertex for all elements
that share this vertex.
89
Field term
b b ,
b ,
jj jG T
j j j jT
dxdy N D d d
N D d d b
with jE A b b b
bj
93
Numerical task of FEM
0 bA
Solve system
of N linear equations where N is the
number of vertices.
Matrix A and vector b only depend on the
triangulation and on the basis functions and
the unknowns are the coefficients = (φi) .
94
FEM
The connection between the elements gives
off-diagonal terms in the matrix A.
Finally one must also include the boundary
terms, which appear as before on the
right side of the equation.
Applethttp://www.lnm.mw.tum.de/teaching/tmapplets/
95
Stresses in a hinge
96
Stresses in a clip
97
Network of trusses
98
Time dependent PDE‘s
2 1, , ,
Tx t T x t W x t
t C C
Simple example is heat equation:
T is temperature, C is specific heat
ρ is density, κ is thermal conductivity
and W are external sources or sinks.
99
Time dependent PDE‘s
1 12
1 1 4
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
ij ij i j i j
ij ij ij ij
tT x t t T x t T x t T x t
C x
tT x t T x t T x t W x t
C
„line method“ in two dimensions:
2
1
4
t
C x
clearly unstable if
Courant-Friedrichs-Lewy (CFL) condition (1928)
Unstable 1d parabolic PDE
100
101
Crank - Nicolson method
2 2
2
2
( , ) ( , ) ( , ) ( , )
( , ) ( , )
tT x t t T x t T x t T x t t
C
tW x t W x t t
C
21( ) ( , ) , ( ) ( , ) , , ...,n nT t T x t W t W x t n L
implicit algorithm
define
Phyllis
Nicolson
John
Crank(1947)
102
Crank - Nicolson method
1 12
4
O ( , ) ( , ) ( , )
( , ) ( , ) ( , )
n n n
n L n L n
tT x t T x t T x t
C x
T x t T x t T x t
Define operator O
Then Crank – Nicolson becomes:
1
2
2
( , ) ( , ) O ( , ) O ( , )
( , ) ( , )
T x t t T x t T x t T x t t
tW x t W x t t
C
103
Crank - Nicolson method
2 2
O ( ) O ( ) ( ) ( )t
T t t T t W t W t tC
1 1
Then Crank – Nicolson becomes:
where 1 is the unity operator.
2 2( , ) ( , ) O ( , ) O ( , )
( , ) ( , )
T x t t T x t T x t T x t t
tW x t W x t t
C
104
Crank - Nicolson method
Calculate the inverted operator B before:
2
2
-1 O
( ) O ( ) ( ) ( )t
T t t T t W t W t tC
B 1
B 1
105
Crank - Nicolson method
Example: 1d diffusion equation: 2
2
u u
Dt x
1 12
1 1
22
2
i i
i i i
i i i
u t t u t
tD
u t t u t t u t tx
u t u t u t
Crank-Nicholson discretization:
22
D t
x
Courant-Friedrichs-Lewy (CFL) number
1 1
1 1
1 2
1 2
i i i
i i i
u t t u t t u t t
u t u t u t
1 12
1 1
22
2
i i
i i i
i i i
u t t u t
tD
u t t u t t u t tx
u t u t u t tridiagonal problem
106
Tridiagonal matrix problem
equation:
modify coefficients:
solution:
Algorithm goes like O(N) (instead of O(N3) in Gauss elimination).
107
Wave equation2
2 22
with
y kc y ct
2 21 12
2( , ) ( , ) ( , ) ( , )n k n k n k
n k
y x t y x t y x tc y x t
t
21 1
21 1
2 1 2
( , ) ( ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
n k n k n k
n k n k n L k n L k
y x t y x t y x t
y x t y x t y x t y x t
with = cΔt/Δx < 1/ √ 2
which corresponds to
cut off modes for wave
lengths smaller than .
−
108
InitializationTo start the iterations one needs to know
the field at two times t and t-Δt.
That means, one needs to know y(xn,0) and 0
,n
yx
tSet 0 0
, , ,n n n
yy x t y x t x
t error O(Δt)
2
2
1 1
1 0 0
0 0 0 04
, , ,
, , , ,
n n n
n n n L n L
yy x t y x t x
t
y x y x y x y x
better
error O(Δt2)
109
Solution of the wave equation
110
velocity field, pressure field
Navier – Stokes equation
10
1
1
2
2
( ) ,
Euler: ( )
Stokes:
vv v p v v
t
vv v p
t
vp v
t
( , ) ( , )v x t p x t
equation of motion for incompressible fluid
viscosity
Solvers for NS equation
• Penalty method with MAC
• Finite Volume Method (FLUENT, OpenFOAM)
• Spectral method
• Lattice Boltzmann (Ladd)
• Discrete methods: DPD, SPH, SRD, LGA,…
• k-ε model for turbulence
CFD = Computational Fluid Dynamics
113
Navier – Stokes equation
211 ( )k k
k k k k
v vp v v v
t
1 0k kv v
2 211 k k
k k k k
v vp v v v
t
Insert incompressibility condition:
Apply on both sides :
114
Navier – Stokes equation
Poisson equation determine pressure pk+1
21 ( ) k k kp v v
To solve it, one needs boundary conditions
for the pressure which one obtains projecting
the NS equation on the boundary.
This must be done numerically.
115
Operator splitting
211
* *
( ) k kk k k k
v v v vp v v v
t
2
11
*
*
( )
kk k k
kk
v vv v v
t
v vp
t
Introduce auxiliary variable field v*
and split
in two
equations: v*
116
Operator splitting
21
*
k
vp
t
11 1
1 * kk k
pn p n v v
n t
11
*k
k
v vp
t
Applying on
one obtains
Projecting on the normal n to the boundary
one obtains:
117
Spatial discretization
MAC = Marker and Cell is a staggered lattice:
Place components of velocity on middle of
edges and pressures in the centers of the cells.
h is the
lattice
spacingy
x
118
Spatial discretization
11 2
1
, ,, , hi j i jx i j
p p p
21 1 1 12
14 , , , , , ,
hi j i j i j i j i j i jp p p p p p
y
x
○
119
Spatial discretization
1 1 1 1
2 2 2 2
1
* * * * *,
, , , , , , , ,hi j
x i j x i j y i j y i jv v v v v
Poisson equation for the pressure pk+1
is solved on the centers of the cells () .
21
*
k
vp
t
y
x
○
120
Spatial discretization
x x x y xv v v x v v y v
21 1 ( )k k k k k kv v t p v v v
The equations for the
velocity components
are solved on the edges.
○
121
Spatial discretization
1 3 11
2 2 22
1
2, , ,,
h
xx x x x
i j i j i ji j
vv v v v
x
1 1 1 11
2 2 2 22
1 1
2 2
1 1
1 1
1
4
1
2
, , , ,,
, ,
h
xy y y y y
i j i j i j i ji j
x x
i j i j
vv v v v v
y
v v
y
x
○
122
Flow around a vocal chord
123
Sedimentation
comparing experiment and simulation
Glass beadsdescendingin silicon oil
124
Finite Volume MethodR.J. LeVeque, «Finite Volume Methods for Hyperbolic
Problems» (Cambridge Univ. Press, 2002)
, , ,v x t f v x t g v x t
t
i iG G
vf v dV g v dV
t
Solve conservation law
in integral form
i i iG G G
vdV f v n dS g v dV
t
using Green’s theorem:
125
Finite Volume Method
0
i iG G
vdV f v n dS
t
change of value
in volume i 1
i
i
Gi
vf v n dS
t G
126
Forward-Time Central-Space (FTCS)
2
2
2
2
, , ,
( ) ( ), , ,i i
v vf v x t
t x
v t t v t vf v i t
t x
f is spatially discretized in a central difference scheme
1 12
( ) ( ) ( ( )) ( ( ))i i i i
tv t t v t f v t f v t
x
127
FTCSTime evolution of the inviscid Euler equation
using a forward time central space scheme
128
Kurt Friedrichs
Lax-Friedrichs Scheme
Peter Lax
1 1 1 1 2
1
4
( )
( ) ( ) ( ) ( ) ( )
i
i j i j ij ij nn nnnn
u t t
tu t u t u t u t H t S
x
1 1 1 1
1
2 2
( )
( ) ( ) ( ( )) ( ( ))
i
i i i i
v t t
tv t v t f v t f v t
x
129
Lax-Friedrichs Scheme2d Euler equation with reflecting boundaries
FLUENT
130
FLUENT
131
FLUENT
132
133
Cavity with FLUENT
134
Turbine with FLUENT
135
Airfoil with FLUENT
136
Shock wavesSolutions of
parabolic equations which move with constant velocity
and develop a sharp front.
example: tsunami
typical initial condition:
Riemann problem
137
Shock waves
138
Godunov Scheme
Sergei K. Godunov (1959)
Example
1d inviscid Burgers equation: 0
t x
, ,i i i i i it t t t t t t t t
t
xF F
in-flow out-flow
2
if > 0if 0
if < 0if 02
0 if 0 <
, , ,
L LL
L R R RR
L R
gF g g
with
L R L R
2 L R
139
1d Burgers equationformation of shock wave
140
Spectral MethodsSteve Orszag
(1968)
Finite elements:
basis functions: local smooth functions
Spectral methods:
basis functions: global smooth functions
PDE solver for smooth
solutions without adaptive meshing.
Has excellent convergence properties.
141
0 0
with and
( , ) ( ( , ))
( , ) ( , ) ( )B I
Lu x t f u x t
u t u u x u xPDE:
1 1
, ,N
i i N i ii i
u x t a t x u x t a t x
Expand in terms of basis functions ϕi:
Spectral Methods
e.g. ( , ) ( , )Lu x t u x t
t xL differential operator
142
0
0 1 ( , ) ( ( , )) , , ...,L
jLu x t f u x t w x dxdt j N
Define N (orthogonal) test functions wj (x):
wj (x) = ϕj (x) is called the Galerkin method
and wj (x) = δ(x-xj) is called a collocation.
Spectral Methods
143
Spectral Methods
0 0 2
on ( , )
u u
t x
Example 1: 1d advection equation
truncated expansion:
2
2
( ) ( , ) ( ) ( )N
Nl l
l N
u x t a t x
144
Spectral Methods
trigonometric basis and test functions:
1
2
and ( ) ( )ilx ikx
l kx e w x e
2 2
20
10
2
( ) N
ilx ikxl
l N
a t e e dxt x
02 2
, , ...,kk
da N Nika k
dt
2
0
2
( )i l k xlke dx
145
Spectral Methods
0 kk
daika
dt
2
0
0
( ) ( ) kxk Ia u x e dx
solve initial condition
with
( ) sin cos( )Iu x xchoose for instance
2
( ) sin ( ) iktk k
ka t J e
Bessel function
146
Spectral Methods
2
( ) sin ( ) iktk k
ka t J e
From asymptotic behaviour of Bessel functions:
0 for : ( ) pkp k a t k
2
2
( ) ( , ) ( ) N
N ikxk
k N
u x t a t econverges faster
than any
power of 1/N.
147
Spectral Methods
Example 2: 1d (full) Burgers equation
t x xxu u u u
2
2
( ) ( , ) ( ) N
N ikxk
k N
u x t a t e
, , ,t x xxu w u u w u w2
0
, ( ) ( )f w f x w x dx
integral or «weak» form,
with
, :w t
2 2 ( ) , , ...,ikx N N
w x e k
Fourier-Galerkian expansion
148
Spectral Methods
, , ,t x xxu w u u w u w
21
2
, ,ikx ikx
t x xu e u u e
2 21 1
2 2 , , ,ikx ikx ikx
t x x xu e u u e u u ike
integrating by parts:
149
Spectral Methods
2
0
2
( ),ilx ikx i l k xlke e e dxuse orthogonality relation
to solve
2
2
2
, ( ) , N
ikx ilx ikxt t l t k
l N
u e a t e e a
2 22
2 2
2
1 1
2 2
22
( )
,
( )
,
, ,
, ,
N Nikx i l m x ilx ikx
x k l ll m N l N
i l m x ikx ilx ikxk l l m l k
l m l l m k
u u ike a a e i l a e ike
ika a e e k l a e e i k a a k a
21
2 , ,ikx ikx
t xu e u u ike
150
Spectral Methods
22 2
t k k l kl m k
a i k a a k a
2
2
( ) ( ) ( ) ( )kk l k
l m k
a ikt a t a t k a t
t
This system of coupled ODE can be solved e.g. with
Runge Kutta using the Fourier transformed initial
condition: 2
0
1 10 0 0
2 2
( ) ( , ), ( , )ikx ikx
ka u x e u x e dx
151
Spectral Methods with other basis functions
Families of orthogonal polynomials on [-1,1] are Legendre and Chebychev polynomials.
Fourier decomposition is good when functions are periodic.
Laguerre polynomials on [0,∞)
Hermite polynomials on (-∞, ∞)
152
Discrete fluid solvers
• Lattice Gas Automata (LGA)
• Lattice Boltzmann Method (LBM)
• Dissipative Particle Dynamics (DPD)
• Smooth Particle Hydrodynamics (SPH)
• Stochastic Rotation Dynamics (SRD)
• Direct Simulation Monte Carlo (DSMC)
153
Lattice gas Automata
• D.H. Rothman and S. Zaleski, „Lattice-Gas Cellular Automata“ (Cambridge Univ. Press, 1997)
• J.-P. Rivet and J.P. Boon, „Lattice Gas Hydrodynamics“ (Cambridge Univ. Press, 2001)
• D.A. Wolf-Gladrow, „Lattice-Gas Cellular Automata and Lattice Boltzmann Models“ (Lecture Notes, Springer, 2000)
154
Lattice gas AutomataParticles move on a triangular lattice and
follow the following collision rules:
Momentum is conserved at each collision.
It can be proven (Chapman-Enskog) that
its continuum limit is the Navier Stokes eq.
Lattice gas Automata
157
von Karman street
velocity field of a fluid behind an obstacle
Each vector is an average over time of the
velocities inside a square cell of 25 triangles.
159
Lattice gas Automata
Problem in three dimensions, because there exists
no translationally invariant lattice which is
locally isotropic. One must study the model in 4d
and then project down to 3d. Start with 4d
face centered hypercube that has 24 directions
giving 224 = 1677216 possible states. Projecting onto
a 3d hyperplane that already contains 12 directions
adds another six new directions giving 18 in 3d.
160
Discrete fluid solvers
• Lattice Gas Automata (LGA)
• Lattice Boltzmann Method (LBM)
• Dissipative Particle Dynamics (DPD)
• Smooth Particle Hydrodynamics (SPH)
• Stochastic Rotation Dynamics (SRD)
• Direct Simulation Monte Carlo (DSMC)
161
From LGCA to Lattice Boltzmann Models (LBM)
• (Boolean) molecules to (discrete) distributions
ni fi = < ni >
• (Lattice) Boltzmann equations (LBE)
( , 1) ( , )ii i if x c t f x t C f
Lattice Boltzmann
S.Succi, The Lattice Boltzmann equation for fluid dynamics and beyond, Oxford Univ. Press, 2001
ni is the number of particles in a cell going in direction i
162
Boltzmann equation
, ,f x v t x v distribution function
is the number of
particles having at time t velocities
between v and v + Δv in the elementary
volume between x and x + Δx.
, , , , t x vf x x v v t t f x v t t f x f v f
0
, , , ,
lim t x vt
f x x v v t t f x v tf v f a f
t
0
lim
t
va
t
Taylor expansion:
Ludwig Boltzmann
163
Boltzmann equation
Due to collisions between particles in the
volume Δx during the time interval Δt
some additional particles
acquire velocities between v and v+Δv and
some particles do not anymore
have velocities between v and v+Δv , giving the
collision
term:
, ,collf x v t
, ,collf x v t
, , , ,coll coll collf x v t f x v t
164
Boltzmann equation
This gives the Boltzmann equation:
t x v collf v f a f
In thermal equilibrium one expects
the Maxwell-Boltzmann distribution:
22
2
eq kT mn
v u
kTf e
( , )u x t
165
BGK collision term
P.L. BhatnagarBGK model:
P.L. Bhatnagar, E.P. Gross and M. Krook (1954)
eq
coll
f f
where τ is a relaxation time
2
s
m
kT ccs is «sound speed»
μ is viscosity 2 s
kTc
m
166
Averaged quantitiesMoments of the velocity distribution:
, , ,x t m f x v t dv
2
2
( )
, , , ,v u
x t e x t m f x v t dv
, , , ,x t u x t m v f x v t dv
mass density:
momentum density:
energy density:
168
Knudsen numberValidity of the continuum description:
characteristic length of system L must be much larger
than the mean free path l of the molecules
(distance between two subsequent collisions).
K l L
Navier-Stokes equation: 0.01 > K
Boltzmann equation: 0.005 > K
170
Chapman-Enskog expansion0
( )n n
n
f K f
where the small parameter K is the Knudsen number
0 ( ) eqf f
1 1
( )( )
, n n nx x n
n n
K Kt t
Chapman-Enskog
175
1 11
( ) ( )
( ) x xu u u e at
Chapman-Enskog
0 11 1 1 1 2 0 2 0 2
2 1
1
( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) x v x v
f fv f a f v f a f f
t t
0 11 1 1 1 2
2 1
1
( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( ) x v
f fv f a f f
t t
110 1
2
( ) , eq
xy
uv v f f dv
t
Navier Stokes equation:
momentum conservation
176
Gaussian quadrature theorem
0
( ) ( ) ( )b n
i iia
g x w x dx w g x
0
( ) , , ...,b n
ki
k ì i ka
x xw w x dx i n
x x
Be g(x) a polynomial of at most degree 2n+1
if for the positive weight function w(x) there exists a
polynomial p(x) of
degree n+1 such that 0 0 ( ) ( ) , , ...,b
k
a
x p x w x dx k n
with
and xi , i = 0,...,n are the zeros of p(x).
177
Lattice Boltzmann
22 2
2
2 2 4 22
2 12 22
eq sd
s s ss
vc vuvu u
fc c cm c
e
2 with ss
u kTc
c m
small parameter:
w(x) p(x)2
0
( )i ii
a H v
2
2
2
2
d
eq kT m
v u
m kT mf e
178
Hermite Polynomials2 3
0 1 2 31 1 3 ( ) , ( ) , ( ) , ( )H x H x x H x x H x x x
2
2
( ) ( ) ! xi j ijH x H x e dx i
179
Lattice Boltzmannone dimensional case:
2
22
2
1
( ) s
s
vcw v
ce
n + 1 = 31 1
03 3
, ,iv
220 1 2
1 1 2 1
6 3 61
, ,
( )!, ,
( ) ( )i
in i
nw
n H v
180
Lattice Boltzmannthree dimensional case:
2 222
2 2 2 22 2 2 2
y zx
s s s s
v vvvc c c ce e e e
0 0 0 0 0 0
0 01 3 0 0 0 1 3 0 0 0 1 3 1 3
01 3 1 3 0 0 1 3 1 3 1 3 0 1 3 1 3 1 3
1 3 1 3 1 3 1 3 1 3 1
8 27
2 27
1 54
( , , )
( / , , ) ( , / , ) ( , , / ) /
( / , / , ) ( , / , / ) ( / , , / ) / /
( / , / , / ) / / /
w w w w
w w w w w w
w w w w w w
w w w w3
1 216
27 discrete velocity vectors
181
Lattice Boltzmann
D2Q9
D3Q15
D3Q19
Lattice Boltzmann
where the equilibrium distribution is defined as:
Define on each site x of a lattice on each outgoing
bond i a velocity distribution function f(x,vi,t)
which is updated as:
2 20
2 4 2
93 31
2 2
i n is s s
vuvu uf w
c c c
011
( , , ) ( , , ) ( ) ( , , ) ( , , )i i i i i i i i n i if x v v t f x v t F v f u T f x v t
183
Lattice Boltzmann
discretization
1
v t
x
CFL number
22
s
t
c
2 2
2
, , , , t tx x v v t t x v t
tf f t v f v f
js is the inverse of a relaxation time.Orthogonal polynomials
Projections of the distribution
Shear viscosity
Bulk viscosity
Chapman-Enskog expansion:
Multi-Relaxation-Time (MRT) LBM
P. Lallemand and L.S. Luo
Phys.Rev.E 61, 6546 (2000)
0
( , ) ( , ) ( )N
j eqj j j
j j j
sf x c t t t f x t m m
i
mj j f
mj (,...,ux,...)
cs2 1
s9,...,13
1
2
59cs2
9
1
s2
1
2
D3Q15
186
Powerflow, EXA
Car design
Lattice Boltzmann
187
Raising of a bubble
188
3d Rayleigh Benard
189
Flow through porous medium in 2d
using a NVidia GTX680
190
Surface Flow with Moving and Deforming Objects
Interfaces and free surfaces
191
192
Discrete fluid solvers
• Lattice Gas Automata (LGA)
• Lattice Boltzmann Method (LBM)
• Dissipative Particle Dynamics (DPD)
• Smooth Particle Hydrodynamics (SPH)
• Stochastic Rotation Dynamics (SRD)
• Direct Simulation Monte Carlo (DSMC)
193
• SPH describes a fluid by replacing its continuum properties with locally (smoothed) quantities at discrete Lagrangian locations meshless
• SPH is based on integral interpolants (Lucy 1977, Gingold & Monaghan 1977, Liu 2003)
(W is the smoothing kernel)
• These can be approximated discretely by a summation interpolant
'd,' ' rrrrr hWAA
j
jN
jjj
mhWAA
1
, rrrr
Smooth Particle Hydrodynamics
194
The kernel (or weighting Function)
• Example: quadratic kernel
1
4
1
2
3, 2
2qq
hhrW
W(r-r’,h)
Compact supportof kernel
WaterParticles
2h
Radius ofinfluence
r
| | , barh
rq rr
Smooth Particle Hydrodynamics
195
• Spatial gradients are approximated using a summation containing the gradient of the chosen kernel function
• Advantages are:– spatial gradients of the data are calculated analytically
– the characteristics of the method can be changed by using a different kernel
ijijj j
ji WA
mA
ijij
jijii Wm . . uuu
Smooth Particle Hydrodynamics
196
Equations of Motion
• Navier-Stokes equations:
• Recast in particle form as:d
d
jiii j ij
j ij
m Wt
vrv
ijj
ijiji Wm
t vvd
d
iijj
iijj
j
i
ij
i Wpp
mt
Fv
22d
d
v.d
d
t
2d 1
d ipt
v
u F
0
d
d
t
mi
Smooth Particle Hydrodynamics
197
Simulation of free surface
198
Simulation of free surface
199
Simulation of free surface
200
Dwarf Galaxy Formation
201
Discrete fluid solvers
• Lattice Gas Automata (LGA)
• Lattice Boltzmann Method (LBM)
• Dissipative Particle Dynamics (DPD)
• Smooth Particle Hydrodynamics (SPH)
• Stochastic Rotation Dynamics (SRD)
• Direct Simulation Monte Carlo (DSMC)
202
Stochastic Rotation Dynamics
Stochastic Rotation Dynamics (SRD)• introduction of representative fluid particles
• collective interaction by rotation of local particle velocities
• very simple dynamics, but recovers hydrodynamics correctly
• Brownian motion is intrinsic
203
Stochastic Rotation Dynamics
Shift grid to impose
Galilean invariance.
Example of two particles in cell:
204
Shear flow
205
One particle in fluid
particlevv
fluid
e.g. pull sphere through fluid
particlev
Γ
no-slip condition:
create shear in fluid : exchange momentum
movingboundary condition
206
Drag force
AdFD
jiij ij
j i
vvp
x x
drag force
stress tensor
η = μ is static viscosity
(Bernoulli‘s principle)
207
Homogeneous flow
Re << 1 Stokes law:
FD = 6π η R v(exact for Re = 0)
R
v
Re >> 1 Newton‘s law: FD = 0.22π R2v2
general drag law:
CD is the drag coefficient
22
Re8 DD CF
R is particle radius, v is relative velocity
208
Drag coefficient CD
Reynolds number Re = Dv/μ
Re
209
Inhomogeneous flow
In velocity or pressure gradients: Lift forcesare perpendicular to the direction of the external flow,
important for wings of airplanes.
when particle rotates: Magnus effectimportant for soccer
lift force:
CL is „lift coefficient“
2v
2LL C A
210
Many particles in fluids
•The fluid velocity field followsthe incompressible NavierStokes equations.
• Many industrial processesinvolve the transport of solidparticles suspended in a fluid.The particles can be sand,colloids, polymers, etc.
•The particles are dragged bythe fluid with a force:
simulating particles moving in a sheared fluid
22
Re8 DD CF
211
Stokes limit
hydrodynamic interaction between the particles
ij
jjiiji vrrMv
matrixmobility
)(
for Re = 0 mobility matrix exact
Stokesian Dynamics (Brady and Bossis)
invert a full matrix only a few thousand particles
212
Numerical techniques
Calculate stress tensor directly by evaluating the gradients of the velocity field
through interpolation on the numerical grid,e.g. using Chebychev polynomials .
Method of Fogelson and Peskin:Advect markers that were placed in the particle and then put springs between
their new an their old position.These springs then pull the particle.
1
2
Numerical techniques
2 Method of A.L. Fogelson and C.S. Peskin:
Advect markers that were placed in the
particle and then put springs between
their new an their old position.
These springs then pull the particle.
216
Sedimentation
comparing experiment and simulation
Glass beadsdescendingin silicon oil
Sedimentation of platelets
Oblate ellipsoids descend
in a fluid under the action
of gravity.
This has applications inbiology (blood), industry(paint) and geology (clay).
Thesis of Frank Fonseca
θ = 0.15 in 3d
228
Oral exams
Jan.22-Feb.02
2017
229
15 relevant questions
• Congruential and lagged-Fibbonacci RN• Definition of percolation• Fractal dimension and sand-box method• Hoshen-Kopelman algorithm• Finite size scaling• Integration with Monte Carlo• Detailed balance and MR2T2
• Ising model
230
15 relevant questions
• Simulate random walk
• Euler method
• 2nd order Runge-Kutta
• 2nd order predictor-corrector
• Jacobi and Gauss-Seidel relaxation
• Gradient methods
• Strategy of finite elements, finite volumes and spectral methods
231
Next semester402-0810 Computational Quantum Physics
Giuseppe Carleo and Philippe de Forcrand
Tuesday afternoon: V Di 14-16, U Di 16-18
402-0812 Computational Statistical Physics
Mirko Lukovic and Miller Mendoza
Friday morning: V Fr 11-13, U Fr 9-11
327-5102 Molecular Materials Modelling
Daniele Passerone
Friday afternoon: V Fr 14-16, U Fr 16-18
232
Computational Quantum Physics
Giuseppe Carleo and Philippe de Forcrand
Tuesday afternoon: V Di 14-16, U Di 16-18
One particle quantum mechanics:
scattering problem, time evolution
shooting technique
Numerov algorithm
233
Computational Quantum Physics
Many particle systems:
Fock space, etc (≈ 2 weeks theory)
Hartree-Fock approximation
density functional theory and
electron structure (He & H2)
strongly correlated electrons
Hubbard and T-J models
234
Computational Quantum Physics
Lanczos method
Path integral Monte Carlo
Bosonic world lines
QCD, lattice gauge theory
Fermions, QFT
235
Molecular Materials Modelling
Daniele Passerone
Friday afternoon; V Fr 14-16, U Fr 16-18
Empirical potentials and transition rates
Bio-force fields, charges, peptides
Embedded atom models, Wilff‘s theorem
Pair-correlation function with MD
for neutron scattering
236
Melting temperature from phase coexistence
MO-theory, basic SCF, chemical reactions
Density functional theory, pseudopotentials
DFT on realistic systems, hybrids
Linear scaling, GPW
Electronic spectroscopies, STM
Bandstructure, graphene, free energies
Molecular Materials Modelling
237
Computational Statistical Physics
Mirko Lukovic and Miller Mendoza
Friday morning: V Fr 11-13, U Fr 9-11
Advanced Monte Carlo techniques:
continuous variables (XY, Heisenberg)
multi-spin coding, bit-manipulation
vectorization, parallelization
histogram methods, multi canonical
238
Computational Statistical Physics
Kawasaki dynamics, heat bath
microcanonical, Creutz algorithm, Q2R
critical slowing down, dynamical scaling
cluster algorithms (Swendsen-Wang, Wolff)
Monte Carlo Renormalization Group
Molecular Dynamics Simulations:
Verlet and leap frog methods
linked cell method, Verlet tables
239
Computational Statistical Physics
parallelization, realistic potentials
Ewald sums, reaction field method
Nose-Hoover thermostat, rescaling
constant pressure MD, melting
Discrete Elements, friction, inelasticity
rotation and quaternions
ab- initio calculations, Car Parinello