Issue 61

N. Razali et alii, Frattura ed Integrità Strutturale, 61 (2022) 214-229; DOI: 10.3221/IGF-ESIS.61.14

The update for two-step IMR is given by

2         n n n n n n u u u u u u 1 1 4 10 4 16  

2

(12)

The stability function for two-step IMR is

2

z



1

 R z

2



(13)

( )

  2 1  z

4

The two other methods, which are used as a comparison for the symmetriser, are the FTCS square-wave method

 c t





1 u u  n

  n

1  u u  n j

n

(14)



j

j

j

1



x

and the step-wave Lax method

1 2





1    n n n j j j u u u 

(15)

1

  c t

1 2











n

1  u u n j

  n

1  u u  n j

n

1



u

(16)





j

j

j

1

1

x

2

R ESULTS AND D ISCUSSION

T

he method is tested on the advection equation

   

u C u t x

 

(17)

0,

where

1, 1    x C

and the initial condition

0,  

x x

1 0

2

 

  , 0 u x u x   0 



    x x

(18)

0

1

  

x

1

1

2



Results for advection equation: results of FTCS differencing (square-wave) method 1) CFL=0.01

Fig. 1 shows the application of the FTCS differencing (square-wave) method to the problem with CFL=0.01, and no convergence of the exact solution at any time is observed. Fig. 2 shows the results more clearly, in 3D, and with no convergence, but the exact solution is not reached. In the next step, the results are displayed by changing the CFL value and showing whether it attains convergence or reaches the exact solution.

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