Issue 61

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

The stability function for this method is given by

z

 

1 1

2



R z

(5)

( )

z

2

Smoothing for one-step IMR is given follows:

1 2 3 1 4 4 0 1

1 1 2 2 3

(6)

2

These methods are called symmetrisers, are first order and have a stability function given by

  2 2 1 1  z



R z

( )

(7)

The two-step symmetriser is constructed with a composition of four symmetric steps:

A

0 0 0 0 0

     

     

      

     b Pv

c

    

,          

,        

T eb A

 b Pu e c 





, ,    A b c

 

(8)

    e c

u

T

T

2 3

eb eb A

0

 

v

e c

T

T

T

eb eb eb A



This method satisfies B (2) and C (1), which is

2          T T T T b e b c b c b Ac

 2, 2 2 2      T T u c v e c



,

(9)









 2 1           T T T b c u c v e c b c u c v e c 8 8 T 1 2 4 4 T T 

,

.

where ,   A b and  c are given in (8). The stability for this two ‐ step symmetriser is given by   1 ( ) 1 .          T R z zb I zA e

(10)

Using the second ‐ order condition in (9) and satisfying damping, i.e. the stability condition in (10), the two ‐ step IMR with Butcher tableau is obtained as

1 1 2 2 3

1 2 1 1 1 1 1 1 1 2 17 13 3 16 16 16

2 5 2 7 2

(11)

1 2



1

16

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