Hyperbolic tangent stretching functions per M. Vinokur JCP 1983

8/7/2019 Hyperbolic tangent stretching functions per M. Vinokur JCP 1983

http://slidepdf.com/reader/full/hyperbolic-tangent-stretching-functions-per-m-vinokur-jcp-1983 1/6

Some results for the common hyberbolic tangent-based grid stretchingschemes associated with Marcel Vinokur

The one-sided stretching scheme and its analytic version in the limit as ∆ 0.

s∆ _, L_, Ξ _ : 1 Tanh∆ Ξ L 1

Tanh∆s∆, L, Ξ FullSimplify

Limit, ∆ 0

1 Coth∆ Tanh∆ 1 Ξ

L

Ξ

L

Partial derivatives with respect to each variable, including the limiting cases.

Ds∆, L, x, xLimit, ∆ 0∆ Coth∆ Sech 1

x

L ∆2

L

1

L

Ds∆, L, x, ∆Limit, ∆ 0

1 x

L

Coth∆ Sech 1 x

L

∆2

Csch∆2Tanh 1

x

L

∆

0

Ds∆, L, x, LLimit, ∆ 0

x ∆ Coth∆ Sech 1 x

L ∆2

L2

x

L2

Some numerical results for implementation verification

Ns∆, L, Ξ . ∆ 7, L 3, Ξ 1 9, 16 NDs∆, L, Ξ , ∆ . ∆ 7, L 3, Ξ 1 9, 16 N

D

s

∆, L, Ξ

, L

.

∆ 7, L 3, Ξ 1

9

, 16

NDs∆, L, Ξ , Ξ . ∆ 7, L 3, Ξ 1 9, 161.130109424675245 10

6

2.053316509625048 106

4.827691460278840 107

0.00001303476694275287



Ns∆, L, Ξ . ∆ 3, L 2, Ξ 1 11, 16 NDs∆, L, Ξ , ∆ . ∆ 3, L 2, Ξ 1 11, 16 NDs∆, L, Ξ , L . ∆ 3, L 2, Ξ 1 11, 16 NDs∆, L, Ξ , Ξ . ∆ 3, L 2, Ξ 1 11, 160.001553191024452532

0.002512942959289942

0.0008866151849355177

0.01950553406858139

Ns∆, L, Ξ . ∆ 4, L 1, Ξ 1 1000, 16 NDs∆, L, Ξ , ∆ . ∆ 4, L 1, Ξ 1 1000, 16 NDs∆, L, Ξ , L . ∆ 4, L 1, Ξ 1 1000, 16 NDs∆, L, Ξ , Ξ . ∆ 4, L 1, Ξ 1 1000, 16

5.388915112675453 106

9.425182395066187 106

5.410484940505296 106

0.005410484940505296

Ns∆, L, Ξ . ∆ 4, L 1, Ξ 1 100000 000, 16 NDs∆, L, Ξ , ∆ . ∆ 4, L 1, Ξ 1 100000 000, 16 NDs∆, L, Ξ , L . ∆ 4, L 1, Ξ 1 100000 000, 16 NDs∆, L, Ξ , Ξ . ∆ 4, L 1, Ξ 1 100000 000, 16

5.367402865013899 1011

9.392957375935146 1011

5.367403079566020 1011

0.005367403079566020

Limits∆, L, Ξ , ∆ 0 . L 2, Ξ 1 11LimitDs∆, L, Ξ , ∆, ∆ 0 . L 2, Ξ 1 11LimitDs∆, L, Ξ , L, ∆ 0 . L 2, Ξ 1 11LimitDs∆, L, Ξ , Ξ , ∆ 0 . L 2, Ξ 1 11

1

22

0

1

44

1

2

A particular solution useful for checking solves for ∆ satisfying some conditions

2 HyperbolicTangent.nb



Plots2.345678901234567, 2, Ξ , Ξ , 0, 2

Ns2 345 678 901 234 567

1 000 000 000 000 000, 2, 1 2, 16

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

0.04012773950620386

The two-sided stretching scheme and its analytic version in the limit as ∆ 0.

u∆ _, L_, Ξ _ :1

21

Tanh∆ Ξ L 1 2Tanh∆ 2

u∆, L, Ξ FullSimplify

Limit, ∆ 01

2

1 Coth∆

2

Tanh∆ 1

2

Ξ

L

Ξ

L

Partial derivatives with respect to each variable, including the limiting cases.

Du∆, L, x, ∆ FullSimplify

Limit, ∆ 0

Csch ∆

2

2

Sech1

2

x

L ∆

2

L 2 x Sinh∆ L Sinh 1 2 x

L ∆

8 L

0

Du∆, L, x, L FullSimplify

Limit, ∆ 0

x ∆ Coth ∆

2 Sech

1

2

x

L ∆

2

2 L2

x

L2

HyperbolicTangent.nb 3



Limitu∆, L, Ξ , ∆ 0 . ∆ 7, L 3, Ξ 1 9LimitDu∆, L, Ξ , ∆, ∆ 0 . ∆ 7, L 3, Ξ 1 9LimitDu∆, L, Ξ , L, ∆ 0 . ∆ 7, L 3, Ξ 1 9LimitDu∆, L, Ξ , Ξ , ∆ 0 . ∆ 7, L 3, Ξ 1 9

1

27

0

1

81

1

3

A particular solution useful for checking solves for ∆ satisfying some conditions

Plotu6.7890123456789012, 2, Ξ , Ξ , 0, 2

Nu6 7 890 123 456 789 012

10000000 000 000 000, 2, 1 2, 16

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

0.03141345703210717

Visualizing the one- and two-sided schemes

Comparing the two schemes on a non - unit domain

HyperbolicTangent.nb 5



ManipulatePlots∆, 2, Ξ , u∆, 2, Ξ , Ξ , 0, 2, ∆, 3, 1 1000, 25

∆

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

Comparing the scheme' s deviation from lthe limiting ∆ 0case

ManipulatePlots∆, L, Ξ Ξ L,

u∆, L, Ξ Ξ L

, Ξ , 0, L, ∆, 1., 0, 10, 1 25, L, 2, 1, 10

∆

L

0.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

6 HyperbolicTangent.nb

Hyperbolic tangent stretching functions per M. Vinokur JCP 1983

Documents

Transcript of Hyperbolic tangent stretching functions per M. Vinokur JCP 1983