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Table 11 Invariants for development of an undular bore

From: A numerical investigation of the GRLW equation using lumped Galerkin approach with cubic B-spline

Time

\({I_{1}}\)

\({I_{2}}\)

\({I_{3}}\)

pĀ =Ā 2

pĀ =Ā 3

pĀ =Ā 4

pĀ =Ā 2

pĀ =Ā 3

pĀ =Ā 4

pĀ =Ā 2

pĀ =Ā 3

pĀ =Ā 4

Our results for \({U_{0}}=0.1,x_{0}=0,d=5,\mu =1/6,h=0.1,{\Delta{t}}=0.1,x\in \left[ -36,300\right] \)

0

3.5949

3.5949

3.5949

0.3344

0.3344

0.3344

0.0031

0.0031

0.0031

50

3.6051

3.6050

3.6049

0.3348

0.3350

0.3350

0.0019

0.0016

0.0015

100

3.6051

3.6050

3.6050

0.3348

0.3350

0.3350

0.0018

0.0016

0.0015

150

3.6050

3.6050

3.6049

0.3350

0.3349

0.3350

0.0017

0.0016

0.0015

200

3.6050

3.6050

3.6049

0.3354

0.3349

0.3350

0.0012

0.0016

0.0015

Time

\({I_{1}}\)

\({I_{2}}\)

\({I_{3}}\)

pĀ =Ā 2

Ā Ā 

pĀ =Ā 2

Ā Ā 

pĀ =Ā 2

Ā Ā 

QBSC[28] results for \({U_{0}}=0.1,d=5,\mu =3/2,h=0.2,{\Delta{t}}=0.1,x\in \left[ 0,250\right] \)

0

4.0000

Ā Ā 

0.3759

Ā Ā 

0.0025

Ā Ā 

50

4.8507

Ā Ā 

0.4620

Ā Ā 

0.0034

Ā Ā 

100

5.7016

Ā Ā 

0.5480

Ā Ā 

0.0042

Ā Ā 

150

6.5531

Ā Ā 

0.6341

Ā Ā 

0.0051

Ā Ā 

200

7.4055

Ā Ā 

0.7204

Ā Ā 

0.0060

Ā Ā