Aims
Wednesday, 29 August 2012
• Numerical solution of differential equations
• Techniques
• Stability
• Boundary conditions
• Introduction to parallelism
• Applications to solar physics (MHD)
Wednesday, 29 August 2012
Chris Brady
Centre for Fusion, Space and Astrophysics, Warwick
Computer Simulations
in Solar Physics
B)
Wednesday, 29 August 2012
vxn+1
t
vxn
q n
= (vy Bz )
m
dvx
vxn+1 vxn
⇥
dt
t
vx (t)
dx
=v
dt
dvx
vx (t + dt)
= lim
dt 0
dt
dt
Wednesday, 29 August 2012
dv
q
= (v
dt
m
Wednesday, 29 August 2012
Wednesday, 29 August 2012
-10
-5
vy0 = 0
vx0 = qBz/m
0
5
10
t = 0.01 m/(qBz)
x0y = 1
n
xn+1
=
x
y
y +
tvyn
tq n
=
+
(vy Bz )
m
tq n
n
= vy
(vx Bz )
m
n
n
xn+1
=
x
+
tv
x
x
x
vxn
x0x = 0
vyn+1
vxn+1
10
5
0
-5
-10
t2 d2 f (t)
+
2
dt2
t
t
Wednesday, 29 August 2012
0
2
4
6
8
Wednesday, 29 August 2012
0
0.5
1
Time
1.5
d2 V
dV
+
=V
dt2
dt
dV
=V
dt
2
+ O( t3 )
• Ignoring higher order terms
• Effectively diffuses the solution outwards
t) = f (t) +
df (t)
t
dt
Why didn’t it work?
f (t +
Value
t
f (t) = f (t +
)
2
Wednesday, 29 August 2012
f (t +
t df
2 dt
t
t df
t) = f (t +
)+
2
2 dt
Wednesday, 29 August 2012
f (t +
t+
t+
2
t2 d 2 f
+
8 dt2
t
2
t2 d 2 f
+
8 dt2
t
t
t
)
2
2
t
t
t) = f (t +
+
)
2
2
f (t) = f (t +
t+
t+
t
2
t
2
+O( t3 )
+O( t3 )
t+ t/2
⇥
t+
f (t +
t df
2 dt
t+
2
t)
t
f (t)
t2 d 2 f
+
8 dt2
t
2
t2 d 2 f
+
8 dt2
t
t+
t+
t
2
t
2
=
=
=
=
=
n+ 12
vy
xn+1
x
xn+1
y
vxn+1
vyn+1
Wednesday, 29 August 2012
=
n+ 12
vx
+
+
n+ 12
tvy
n+ 12
tvx
+O( t3 )
+O( t3 )
vxn
tq n+ 12
+
(vy Bz )
m
tq n+ 12
n
vy +
(vx Bz )
m
xny
xnx
tq n
(vy Bz )
+
2m
tq n
n
vy +
(vx Bz )
2m
vxn
• Only need to know them to 1st order
timestep
• Need to know RHS of equations at half
df
dt
t
f (t) = f (t +
)
2
Wednesday, 29 August 2012
f (t +
t
t df
t) = f (t +
)+
2
2 dt
Wednesday, 29 August 2012
Wednesday, 29 August 2012
1.5
1
0.5
0
-0.5
-1
-1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Wednesday, 29 August 2012
Wednesday, 29 August 2012
=
k4
=
=
k3
t)
=
k2
f (t +
=
df (t)
dt
k1
=
tg(f (t) +
t
t
k1 , t +
)
2
2
t
t
tg(f (t) +
k2 , t +
)
2
2
tg(f (t) + tk3 , t + t)
k1 + 2k2 + 2k3 + k4
f (t) +
6
tg(f (t), t)
g(f (t), t)
called the Runge-Kutta 4th order algorithm
• Yes. Arbitrarily high order is possible.
• Most common high order time integrator is
Can we do better?
Conclusions
Wednesday, 29 August 2012
integrator instead
• For many particles usually use Boris
cases
• 4th order good enough for most problems
• Can use sympletic integrator in extreme
problem under consideration
• Have to have high enough order for
Wednesday, 29 August 2012
Wednesday, 29 August 2012
Space
Hyperbolic equations
Wednesday, 29 August 2012
Time
place
+ O( x3 ) ⇥
Error terms
x)
ui+1 ui
2 x
x2 2 u
u
3
+
x
+O(
x
)
x x
2
x2 x
Wednesday, 29 August 2012
wrong
• Odd derivative terms are dispersive
• Numerical dispersion - get wave speeds
• Once again ignoring higher order terms
• Even derivative terms are diffusive
• Numerical diffusion - energy in wrong
Wednesday, 29 August 2012
x) u(x
2 x
x) = u(x)
u(x
u
u(x +
=
x x
x) = u(x) +
u(x +
u
x2 2 u
3
x
+
+O(
x
)
x x
2
x2 x
⇥u(x, t)
⇥u(x, t)
+c
=0
⇥t
⇥x
1
x
c
Lx /2
un0 = unnx
unnx +1 = un1
t=
u0i = e
xi
x = Lx /nx
x = [0, Lx ]
⇥2
Wednesday, 29 August 2012
uni = Sin +
n
i
identify how error terms grow from
iteration to iteration
• Write new equation for the errors and
solution S and error terms ε
• Write the numerical solution as an exact
Von Neumann Stability
Analysis
Wednesday, 29 August 2012
un+1
i
n ⇥
n
u
u
i 1
= uni + c t i+1
2 x
Wednesday, 29 August 2012
c t n
(S
2 x i+1
Sin 1 )
G=
=1+
=
=
n
i 1)
+
c t
(
2 x
c t
i sin(k x)
x
x)
eat eik(x
=
n+1
i
n
i
x)
eat eik(x+
=
0
eat eikx
n
i+1
n
i+1
n
i 1
=
n
i
nx /2
c t
(
2 x
eat eikx
+
n
i
=
=
n
i
n+1
i
n
i
n ik x
ie
n
ik x
ie
Sin+1 +
=0 since exact solution
= Sin +
Wednesday, 29 August 2012
n+1
i
n
i+1
n
i 1)
⇥2
c t
1+
sin(k x)
x
Wednesday, 29 August 2012
• Alternatively use upwind difference
Higher order schemes need less diffusion
to be stable.
• Can be stabilized by adding finite diffusion.
approximations are unconditionally
unstable for pure advection (See Baldauf
JCP 227 6638).
• In fact most “centered” finite difference
Stability of advection
schemes
Wednesday, 29 August 2012
exist values of Δt and Δx that make the
scheme stable
• The error will grow without bounds
• This method is unconditionally unstable
• Some methods are conditionally stable. There
than one
• The amplification factor |G| is always larger
|G| = GG =
⇤
Wednesday, 29 August 2012
|G| =
1
2
c t ik
(e
2 x
⇤
G=1+
x2
t
x
(1
e
c t n
(ui+1
2 x
u
ui+1
=
t2 x
un+1
= uni +
i
Wednesday, 29 August 2012
x) = u(x)
u(x
2
x) = u(x) +
u(x +
⇥2u
=0
⇥x2
)
t
t
⇥2
x
2+e
ik x
)
2uni + uni 1 )
⇥2
c t
+
sin(k x)
x
(eik
(uni+1
+ O( x3 )
x2
1
x2
cos(k x))
ik x
uni 1 )
2ui + ui
x2
x2 2 u
u
3
+
x
+O(
x
)
x x
2
x2 x
u
x2 2 u
3
x
+
+O(
x
)
x x
2
x2 x
⇥u
⇥u
+c
⇥t
⇥x
Wednesday, 29 August 2012
⇧
⇧
⇧
⇧
⇧
⇤
Wednesday, 29 August 2012
1
a
0
0
..
.
a
1
a
0
..
.
0
a
1
a
..
.
|G| =
un+1
i
= Rc
1
2
2
0
0
a
1
..
.
1+
x
...
...
...
...
..
.
c
⌃⇧
⌃⇧
⌃⇧
⌃⇧
⌃⇧
⌅⇤
⇥
1
⇥
t 2
un+1
0
un+1
1
un+1
2
un+1
3
..
.
⌃ ⇧
⌃ ⇧
⌃= ⇧
⌃ ⇧
⌃ ⇧
⌅ ⇤
⇥
sin2 (kx)
un0
un1
un2
un3
..
.
⌃
⌃
⌃
⌃
⌃
⌅
⇥
n+1
n+1 ⇥
u
u
i 1
= uni + c t i+1
2 x
c x
x2
t
Wednesday, 29 August 2012
Wednesday, 29 August 2012
u
x2
2
u
x
x
4th
x
⇥
⇥
4th
FTCS
ui+2 + 16ui+1
1
1
2
2
BTCS
ui
+ ui
4th order
30ui + 16ui
12 x2
ui+2 + 8ui+1 8ui
12 x
Higher order in space
Reflecting boundary
Wednesday, 29 August 2012
• Velocity reverses sign
boundary
• Needs to be physically correct
• Density is unchanged at reflecting
for advection equation.
• Can’t do for every problem. Not possible
Wednesday, 29 August 2012
Wednesday, 29 August 2012
boundary conditions
• Have to specify more ghost cells
• Essentially specify higher order derivative
stencils
• Higher order spatial derivatives have wider
Boundary conditions at
high order
Wednesday, 29 August 2012
numerical gradient at the boundary to zero
• Decent first approximation to set the
• Set all incoming characteristics to zero
outgoing
• Set all outgoing characteristics to still be
characteristics.
• To do properly need to consider
Absorbing Boundary
Wednesday, 29 August 2012
Wednesday, 29 August 2012
•
•
•
Nonlinear equations
Quick and efficient.
Copy data from adjacent processor
Same principle as boundary conditions
Processor 1
Processor 0
Wednesday, 29 August 2012
•
•
•
Wednesday, 29 August 2012
•
•
•
Differential calculus breaks!
•
•
Upwinding
Viscosity
Odd thing is that we can continue.
•
The problem is that the solution steepens into a
discontinuity.
Not the timestep.
What went wrong
now?
Timestep can now change while the code runs
Timestep is created by analogy with the linear
advection equation
Non-linear problem so can’t do von Neumann
analysis
x
t=
MAXVAL(u)
u
u
+u
=0
t
x
Wednesday, 29 August 2012
0.6
0.8
1
u 1 u2
+
=0
t
2 x
u
ui
⇥
x x
ui
x
1
x2 2 u
u
+
x
+O( x3 )
2
x x
2
x x
u
x2 2 u
x
+
+O( x3 )
2
x x
2
x x
0.4
u
ui+1 ui
⇥
x x
x
x) = u(x)
u(x
0.2
x) = u(x) +
0
u(x +
Wednesday, 29 August 2012
-0.2
0
0.2
0.4
0.6
0.8
Wednesday, 29 August 2012
-0.2
0
0.2
0.4
0.6
0.8
Wednesday, 29 August 2012
0.0
0.0
0.2
0.4
0.6
0.8
0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
1
1.0
Wednesday, 29 August 2012
Wednesday, 29 August 2012
ui
x
1
Upwind for right traveling objects
u
ui
⇥
x x
Upwind for left traveling objects
u
ui+1 ui
⇥
x x
x
• Not just diffusion
• Godunov’s order barrier theory
• Only schemes of first order are TVD
shock works!
• Take derivative onesided backwards across
Upwinding
⇥
u
dx =
|ui+1
x
i
ui |
Why not 1st order?
Wednesday, 29 August 2012
• Leading order error term is second order
• Large diffusive error
• Works well enough for shocks
• Very poor for smooth solution regions
Wednesday, 29 August 2012
• TVD scheme is monotonicity preserving
• No spurious oscillations
timestep
• TVD is simply that TV decreases every
TV =
⇤
Total Variation
Diminishing (TVD)
Wednesday, 29 August 2012
T
p=
m
⇧⇤
+ ⇤.(⇤V)
⇧t
⇧V
⇤
+ ⇤(V.⇤)V
⇧t
⇧B
⇧t
⇤
d p⇥
1 dt ⇤
Wednesday, 29 August 2012
⇤ ⇥ (V ⇥ B)
0
=
=
.B = 0
B⇥⇤⇥B
⇤p
=
1
0
=
µ
Hydrodynamics & MHD
Wednesday, 29 August 2012
Wednesday, 29 August 2012
1
A
<
A
t
x
cf
Timestep
• Flux constrained transport
• Divergence cleaners
solenoidal condition
• Apply techniques above
• Have to find some mechanism to ensure
MHD
Wednesday, 29 August 2012
Wednesday, 29 August 2012
Nonlinear
Hydrodynamics &
MHD
• Have to use viscosity to stabilize shocks
• Poor choice for shock dominated problems
order
• Job done
• Good for linear wave problems at high
Finite Difference Code
Wednesday, 29 August 2012
Wednesday, 29 August 2012
.f = 0
⇥u ⇥u2 /2
+
=0
⇥t
⇥x
⇥u
+
⇥t
Conservative form
viscosity.
• This corresponds to a real, physical, shock
shock
• Real shocks have entropy jump across
scheme to resolve shock is NOT shock
viscosity
• Viscosity needed to allow finite difference
Brief aside
1
2
1
2
fi+ 12 )
Choices
fi+ 12
ui+1
t(fi
Wednesday, 29 August 2012
• Godunov schemes - Riemann solvers
order reconstruction
• Flux limiters -Weighted average of upwind and high
equivalent to upwinding in finite difference
• Say flux at edge is the same as flux in the centre.
• If always select flux in upwind direction then
Wednesday, 29 August 2012
fi
ui
n
un+1
=
u
i +
i
Conservative form
Wednesday, 29 August 2012
Wednesday, 29 August 2012
Contact discontinuity
Rarefaction
fan
Pressure
Shock
Wednesday, 29 August 2012
Wednesday, 29 August 2012
• Some Riemann-solver less versions
not assumed to be simple piecewise linear
blocks
• Bram van Leer 1979
• Godunov type scheme where states are
Law scheme
• Monotone Upwind for Scalar Conservation
MUSCL Scheme
Lagrangian Remap
Mesh can tangle
for rotational
flows
Mesh is
compressed at
shocks
Mesh follows
fluid flow
Wednesday, 29 August 2012
code
• Good example is Warwick’s own LARE
is shock capturing
• Take Lagrangian step
• Remap back to original grid
• So long as remap using flux limiters scheme
Wednesday, 29 August 2012
•
•
•
Lagrangian Codes
AMR
Wednesday, 29 August 2012
• Only works well for certain problems
computers
• Allows problems to be run on smaller
“interesting” regions
• Adaptive mesh refinement
• Locally adjusts the grid spacing in
Wednesday, 29 August 2012
• Specialised and hard to program
performance
• Good shock performance, good linear wave
remap every step and don’t have to remap
back to original grid
• Arbitrary Lagrangian Eulerian
• Like Lagrangian Remap, but don’t have to
ALE Codes
Wednesday, 29 August 2012
• Structure design
• Parallelism
• Output and visualisation
• User interaction
software is not science related
• Most of the work in developing scientific
Words on developing
codes