APP~NDIX
B
it turns out that the scheme becomes unstable even without shock formation. We cure the instabilities by the method of pseudo-viscosity [Ri67]. Here we introduce the artificial viscosity term:
where a is the coefficient of artificial viscosity, typically chosen to be of order 1.
Then we modify (B.2b) to:
un+1 _ un _ 6-t
(Fn+t _
Fn+t _ nn ) _ 6-t(cn+t _ Gn+t)
j,l - 1,l 6-z j+l,l j-1,l 1,l 6-r j,1+1 j,1-1 (B.2b')
We have checked that the results are not sensitive to the values of a, as long as a> 0.
We handle the boundaries at z
=
0 and r=
0 by starting the grid points half a step away from the origin:r1 = -0.56.r , r2 = 0.56.r , r3 = 1.56.r , ...
z1 = -0.56.z , z2 = 0.56.z , z3 = l.56.z , ... ,
where 6-r and 6-z are the grid sizes in r and z. directions respectively. Here the first grid points are put in there just to ensure the symmetry properties of the fields.
For example,
ulnl , = u2nl ' F1n1 ' , = -F2n1 ' ' G~ , I= G~ ' I '
for the 1st, 3rd, and 4th components, and
for the 2nd components.
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