A comparative study of TVD limters Well known limiters and new ones

Well known limiters and new ones

Friedemann Kemm BTU Cottbus

kemm@math.tu-cottbus.de

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-4 -2 0 2 4 6

TVD-region Sweby-region linear 3rd order Lax-Wendroff Beam-Warming Fromm

0 0.2 0.4 0.6 0.8 1 1.2

-1 -0.5 0 0.5 1

Ultrabee β=2/3 Superbee

Lax-Wendroff can be written as

a correction of first order upwind.

Lax Wendroff

advection

qⁿ⁺¹

 = qⁿ

 − ν + 1 2

ν(1 − ν)

1

rⁿ



− 1

Δqⁿ

−1/2

with

r = Δq_p Δq_don

ν = distance covered in Δt Δ

Beam-Warming can be written as a correction of first order upwind.

Beam Warming

advection

qⁿ⁺¹

 = qⁿ

 − ν + 1 2

ν(1 − ν)

rⁿ



rⁿ



− rⁿ

−1

Δqⁿ

−1/2

with

r = Δq_p Δq_don

ν = distance covered in Δt Δ

Fromm scheme can be written as a correction of first order upwind.

Fromm

advection

qⁿ⁺¹

 = qⁿ

 − ν + 1 2

ν(1 − ν)

(1 + rⁿ

 )/2 rⁿ



− (1 + rⁿ

−1)/2

Δqⁿ

−1/2

with

r = Δq_p Δq_don

ν = distance covered in Δt Δ

TVD-schemes can be written as

a correction of first order upwind.

General Case

advection

qⁿ⁺¹

 = qⁿ

 − ν + 1 2

ν(1 − ν)

φ(rⁿ

 ) rⁿ



− φ(rⁿ

−1)

Δqⁿ

−1/2

with

r = Δq_p Δq_don

ν = distance covered in Δt Δ

The TVD-region is much larger than the Sweby region.

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-6 -4 -2 0 2 4 6

CFL=0.1 TVD

Sweby

-1 0 1 2 3 4 5

-15 -10 -5 0 5 10 15

CFL=0.5 TVD

Sweby

0 5 10 15 20

-20 -10 0 10 20 30

CFL=0.9 TVD

Sweby

− 2

1 − |ν|≤

φ(r)

r − φ(R) ≤ 2

|ν|

−2≤

φ(r)

r − φ(R) ≤ 2

Third order schemes are upwind biased.

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-6 -4 -2 0 2 4 6

CFL=0.1 3rd order

LW BW Fromm

-1 0 1 2 3 4 5

-15 -10 -5 0 5 10 15

CFL=0.5 3rd order

LW BW Fromm

0 5 10 15 20

-20 -10 0 10 20 30

CFL=0.9 3rd order

LW BW Fromm

φ₃(r) =

1 − 1 + |ν| 3

+ 1 + |ν| 3

r φ_LW(r) = 1 φ_BW(r) = r

Limiters might be constructed by sticking to third order as long as possible.

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-6 -4 -2 0 2 4 6 CFL=0.1

θ=1.00 θ=0.75 θ=0.50 MC

-1 0 1 2 3 4 5

-15 -10 -5 0 5 10 15

CFL=0.5 θ=1.00

θ=0.75 θ=0.50 MC

0 5 10 15 20

-20 -10 0 10 20 30 CFL=0.9

θ=1.00 θ=0.75 θ=0.50 MC

φ_θ(r) = minⁿmxⁿ−(1 − θ) 2

|ν|

, φ₃(r)^o,

mxⁿ−(1 − θ) 2 1 − |ν|

r, θ 2

|ν|

r^o, θ 2 1 − |ν|

o

We introduce an adaption of

Superbee type limiters to third order.

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-6 -4 -2 0 2 4 6 CFL=0.1

Ultrabee β=2/3 Superbee

-1 0 1 2 3 4 5

-15 -10 -5 0 5 10 15

CFL=0.5 Ultrabee

β=2/3 Superbee

0 5 10 15 20

-20 -10 0 10 20 30 CFL=0.9

Ultrabee β=2/3 Superbee

φ_β(r) = mxⁿ0,minφ_UB(r), mx{1 + (φ⁰

3 − β/2)(r − 1),1 + (φ⁰

3 + β/2)(r − 1)} ^o

Superpower is closer to the linear third order scheme than Hyperbee.

-1 -0.5 0 0.5 1 1.5 2 2.5 3

-6 -4 -2 0 2 4 6

CFL=0.1 Hyperbee

Superpower van Leer

-1 0 1 2 3 4 5

-15 -10 -5 0 5 10 15

CFL=0.5 Hyperbee

Superpower van Leer

0 5 10 15 20

-20 -10 0 10 20 30

CFL=0.9 Hyperbee

Superpower van Leer

φ_sp(r) = mxⁿ0,φ₃(r)1 −

1 − |r| 1 + |r|

p(r)o

p(r) = ( ₂

|ν| ·2(1 − φ⁰

3) , r ≤ 1

2

|1−ν| · 2φ⁰

3 , r ≥ 1

Third order does not guarantee for a good representation of the amplitude.

0 0.5 1

-1 -0.5 0 0.5 1

CFL=0.1

θ=0.50 θ=1.00 MC

0 0.5 1

-1 -0.5 0 0.5 1

CFL=0.9

θ=0.50 θ=1.00 MC

Standard example with 200 cells after 10 full rounds (t = 20)

Adapting Ultrabee to third order reduces squaring without sacrificing the amplitude.

0 0.5 1

-1 -0.5 0 0.5 1

CFL=0.1

Ultrabee β=2/3 Superbee

0 0.5 1

-1 -0.5 0 0.5 1

CFL=0.9

Ultrabee β=2/3 Superbee

Standard example with 200 cells after 10 full rounds (t = 20)

On coarse grids the error is mainly due to the amplitude.

0.1 1

100 200 400

CFL=0.1

Superbee Ultrabee β=2/3 Superpower θ=1

0.1 1

100 200 400

CFL=0.9

Superbee Ultrabee β=2/3 Superpower θ=1

L¹-norm of the error

The squaring effect spoils the

convergence of Superbee type limiters.

0.01 0.1 1

100 1000 10000 100000

CFL=0.1

Superbee Ultrabee β=2/3 Superpower θ=1

0.01 0.1 1

100 1000 10000 100000

CFL=0.9

Superbee Ultrabee β=2/3 Superpower θ=1

L¹-norm of the error

Smooth limiters are a

good choice for nonlinear waves.

3 3.5 4 4.5

0 0.5 1 1.5 2 2.5 3

reference solution mixed Superpower and Ultrabee

3 3.5 4 4.5

0 0.5 1 1.5 2 2.5 3

reference solution mixed β=2/3 and Ultrabee

Detail of Shu-Osher problem, t = 1.8, 400 cells

In summary, CFL-dependent limiters enhance the quality of TVD-schemes.

0 1 2 3 4 5 6 7

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

Superbee, primitive Minmod, characteristic Superbee, characteristic Ultrabee, characteristic

Detail of Toro’s test case 3 ¹⁵