@Kitonum 

In case that I have more than one list the "do" loop does not seem to work:

L[1]:=[x1,x2,x3,x4]:
L[2]:=[y1,y2,y3,y4]
......

for j from 1 to 2 do
K[j]:=(i::integer) -> `if`(irem(i,4)<>0, L[j][irem(i,4)], L[j][4]);
od;

Any idea to make this procedure working for any L[i]?

@Kitonum Thank you a lot!

@Carl Love 

For fixed t=T every time I took the solutions of h(x,T) and u(x,T) and made from these polynomial functions (with cubic splines). If the result was analytic (no errors) the above curves should be zero. Above we see that as the t increases the orange curve moves away from zero (x-axis) while the blue remains the same (if not closer). As I mention I dont know if this is a reliable measure of accuracy of the whole solution but the evergoing growth of the orange curve is a sign that something goes very wrong.

@Preben Alsholm 

I dont know if it is a measure of accuracy but I did the following:

I fix t=1 and I found the values of h(x,1) and u(x,1), then with Splines of order 3 I create the functions hspline(x) and uspline(x). Then I checked if these functions are solutions of the PDE system. The orange is for Maple the blue for Mathematica.

For hspline(x):

For uspline(x):

 I dont think that anyone can come to any conclusion just by that but is a start..

EDIT

The same for t=4

hspline(x):

usplines(x)

I think its obvious what will happen if I look further in time. I use Maple since Maple 7 and it is the first time it disappoints me. Please bulit a more reliable scheme for pde systems.

@Preben Alsholm 

So in this case, Maple cannot be trusted? Unfortunately I don't know Mathematica well enough to find the residuals there..

@Carl Love 

Thank you very much for your efforts!

These differences are very minor to affect the total behavior..The mathematica code results a shock wave which maintains its height and also collides plastically with an other wave.

So if Maple is correct then Mathematica is wrong?
I don't know the exact numerical scheme behind Mathematica's NDSolve (I use mailny Maple) but it states that in a rectangular area it use also finite differences..

P.S. I uploaded the worksheet via dropbox because of its size, it was ~9MB. Sorry for the incovinience.

@Carl Love 

(That's an older account of mine)

You can download the .mw file here

Thank you for your interest !

@acer 

I am using Maple since Maple7 and i am really accustomed to the classic inteface..!

Thnak you for the answer!

Giorgos

Thanx a lot, the job done!

Thanx a lot, the job done!

Thanx a lot friend!

I am an old user (from maple 6) and i am used to Classic worksheet!

the problem is that "xmaple" opens Standard worksheet, "maple" opens Command-line and there is not a maple -cw... there is a "cmaple" in a another folder (not in .bin) but is not execudable,not even by terminal...