1. The Finite Volume Method
Flux Limiters TVD
Ingo Philipp
Computational Astrophysics

2. created nor destroyed in [x1, x2]
Integral Form
impermeable wall x1 x2
flow to the right
flow to the left
substance neither
created nor destroyed in [x1, x2]
mass in [x1, x2] at time t2 > t1 in terms of the total mass at time t1 & the total (integrated)
flux at each boundary during [t1, t2]
integral form of the conservation law!

3. Differential
Form
differential form r(x,t) and v(x,t) are differentiable functions, i.e.
this doesn‘t hold if the density
is discontinuous
The integral form is more fundamental physically
and thus the appropriate representation
integral form continues to be valid
even for discontinuous solutions

4. General Form a dS outflow defines a lost of some substance! outflow
inflow
integral form
differential form:
balance law
differential form

5. The Finite Volume Method xi xi+1 xi-1 xi+1/2 xi-1/2 Vi Ui-1 Ui Ui+1
true for any
integral law is transferred to
small control volumes
vertex centered
with
piecewise constant cell average for each xN x1 U
stationary mesh – constant Dx
integrate over small time step Dt – how does the cell mean evolve in time?
mean evolution equation without approximation

6. quadrature – mid-point rule with
The Finite
Volume Method
(i,j)
xi+1/2
xi-1/2
yj-1/2
yj+1/2
(0,1)
(0,-1)
(1,0)
(-1,0) Dx Dy
2D & flux approximation
quadrature – mid-point rule with
1D & without sources Q
the true flux at the interfaces is replaced by a numerical flux function based on

7. Piecewise Linear Reconstruction
use the cell averages to compute a polynomial representation of U for each cell
the easy way out:
polynomial of 0th order
we could instead assume a linear behavior for
for xi
xi-1
xi+1
where
the average value of over the control
volume is regardless of the slope

8. The Advection Equation solve with IC and
new set of variables and gives with x t
characteristic
profile doesn‘t change in shape – it shifts in positive v>0 or
in negative v<0 direction x t tn x t
characteristic
DOD for U(x,t) is just the
single point (x-v(t-tn), tn)

9. The Advection Equation characteristics tn tn+1 xi-1 xi xi+1
since we know the analytical solution we are able to compute the flux integrals
(numerical flux functions) with the help of the polynomial reconstructed , i.e.
and
characteristics tn
tn+1
xi-1 xi
xi+1
for
outflow – backtrack into ith cell at the nth time level
inflow – backtrack into (i-1)th cell at the nth time level
characteristics

10. The Advection
Equation tn
tn+1
xi-1 xi
xi+1

11. Choice of Slopes upwind (Godunov‘s method) centered slope (Fromm)
upwind slope (Beam-Warming)
downwind slope (Lax-Wendroff)
numerical DOD contains physical DOD & von Neumann stable if

12. Upwind artificial diffusion upwind (Godunov‘s method) U(x,t) x
numerical diffusion
local discretization error

13. Lax Wendroff artificial diffusion downwind slope (Lax-Wendroff) U(x,t)
numerical dispersion
local discretization error

14. Beam Warming upwind slope (Beam-Warming) U(x,t) x numerical dispersion
local discretization error

15. Beam Warming upwind slope (Beam-Warming) U(x,t) x
periodic boundary condition
U(x,t) x

16. Fromm centered slope (Fromm) U(x,t) x numerical dissipation
local discretization error

17. ? What went wrong downwind slope (Lax-Wendroff) applied to 1
initial profile
J-1 J
J+1

18. What went
wrong ?
downwind slope (Lax-Wendroff) applied to 1
J-1 J
J+1

19. ? What went wrong downwind slope (Lax-Wendroff) applied to 1 J-1 J J+1
J-1 J
J+1
tn+2
tn+1 tn
J-1 J
J+1 J
J+1

20. ? What went wrong downwind slope (Lax-Wendroff) applied to overshoot
1.125 1
overshoot
0.375
J-1 J
J+1
tn+2
tn+1 tn
J-1 J
J+1 J
J+1

21. ? What went wrong downwind slope (Lax-Wendroff) applied to 1.172 1
0.98
0.7
0.14
initial profile
J-1 J
J+1
overshoot
overshoot
overshoot
undershoot

22. ? What went wrong TOTAL VARIATION
any negative slope in the Jth cell leads to a volume average > 1 at tn+1
to avoid oscillations just set the slope to zero
gives 1st order upwind method
but in smooth regions we want 2nd order accuracy (Lax-Wendroff)
benefit from both
near a discontinuity we may want to limit the slope
in smooth regions we choose sth. like the Lax-Wendroff slope
…how much should we limit the slope? …how to control the flux?
…how do we measure oscillations in the solution?
TOTAL VARIATION

23. Flux Limiter xi xi+1 xi-1 xi-2 …how to control the flux?
the time averaged flux at the interface should now
be determined by the jump xi
xi+1
xi-1
xi-2
gives us a jump
in smooth regions and
limited version of the jump
far from 1 near a discontinuity
we might want a flux limiter f function that
has values near 1 for q~1, but that reduces
or increases the slope where the data is not
smooth
flux limiter function
for
for
measure of the smoothness of the data near

24. Flux Limiter upwind (Godunov‘s method) centered slope (Fromm)
upwind slope (Beam-Warming)
downwind slope (Lax-Wendroff)

25. monotone schemes can be at most 1st order accurate
*High Resolution Schemes for Hyperbolic Conservation Laws
Total
Variation
How does f(x) vary on [a,b] ?
supremum of sums over all partitions
to avoid oscillations we require that the method
doesn‘t increase the total variation (TVNI) 1
for any starting data
Amiram Harten* (1983) p 2p
a monotone scheme is TVNI
if initial condition is then -1
a TVNI scheme is monotonicity preserving
Godunov‘s theorem
monotone schemes can be at most 1st order accurate

26. Harten’s Theorem may in general be data dependent
THEOREM For any scheme of the above form, a sufficient condition for the scheme
to be TVNI is that the coefficients satisfy
advection equation
CFL
for all values of and
and
if we are at an extremum and we should take
Osher and Chakravarthy (1984)
TVD schemes must degenerate to 1st order accuracy at extremal points

27. Total Variation TVNI 2nd order TVNI and P.K. Sweby* (1984) Fromm Fromm
* High resolution schemes using flux-limiters for hyperbolic conservation laws
Total
Variation
and
P.K. Sweby* (1984) 1 2 3
Godunov
Fromm
Beam-Warming
Lax-Wendroff
TVNI 1 2 3
Godunov
Fromm
Beam-Warming
Lax-Wendroff
2nd order TVNI
none of these linear limiters generate a TVNI scheme
any 2nd order scheme relying on must be a weighted average of the
LW and BW scheme

28. MinMod 2nd order TVNI 1 2 3 minmod U(x,t) x slope limiter version
Beam-Warming
upwind slope
Lax-Wendroff
downwind slope
Godunov‘s method
upwind

29. Monotonized Central Difference 2nd order TVNI MC 2 3 1 U(x,t) x
slope limiter version
Godunov‘s method
upwind
Fromm
centered slope
~Beam-Warming
upwind slope
~Lax-Wendroff
downwind slope

30. References

31. Upwind & CFL tn tn+1 tn tn+1 updating scheme = upwind
information travels less than one grid cell
in one time step
information travels more than one grid cell
in one time step
upwind method certainly unstable!
necessary CFL stability condition
fulfilled

32. Numerical Solution upwind (Godunov‘s method) centered slope (Fromm)
U(x,t)
upwind slope (Beam-Warming)
downwind slope (Lax-Wendroff) x

33. Lax Wendroff downwind slope (Lax-Wendroff) U(x,t) x
periodic boundary condition
U(x,t) x