With this:
Tt = 3; L = 0.1; W0 = 2/1000000(*.00002*); U0 = 1; R =
1322/10000 (*13.11*); px = 0(*0.13*); b =
1/100000; eq = {D[W[x, y, t], t] + u[x, y, t] D[W[x, y, t], x] +
Integrate[W[x, y, t], {y, 0, y}]*D[W[x, y, t], y] -
2*R*y*D[W[x, y, t], y] - R*(1 + y^2)*D[W[x, y, t], y, y] -
D[W[x, y, t], x, x] == (y/(1 + y^2))*
Integrate[W[x, y, t], {y, 0, y}] + b*y/(1 + y^2),
D[u[x, y, t], t] + u[x, y, t] D[u[x, y, t], x] +
Integrate[W[x, y, t], {y, 0, y}]*D[u[x, y, t], y] -
R*y*D[u[x, y, t], y] - R*(1 + y^2)*D[u[x, y, t], y, y] -
D[u[x, y, t], x, x] + px == 0};
ic = {W[x, y, 0] == W0*(y - L),
u[x, y, 0] == U0*y/L}; bc = {W[x, L, t] == 0, W[x, 0, t] == -W0*L,
W[0, y, t] == W0*(y - L), u[x, 0, t] == 0, u[x, L, t] == U0,
u[0, y, t] == U0*y/L}; bc1 = {Derivative[1, 0, 0][u][L, y, t] == 0,
Derivative[1, 0, 0][W][L, y, t] == 0};
sol = NDSolve[{eq, ic, bc, bc1}, {W, u}, {x, 0, L}, {y, 0, L}, {t, 0,
Tt}, Method -> "StiffnessSwitching"];
I get different results:
{Plot3D[W[L/2, y, t] /. First[sol], {y, 0, L}, {t, 0, Tt},
PlotRange -> All, AxesLabel -> Automatic, PlotLabel -> W,
Mesh -> None, ColorFunction -> "Rainbow"],
Plot3D[u[L/2, y, t] /. First[sol], {y, 0, L}, {t, 0, Tt},
PlotRange -> All, AxesLabel -> Automatic, PlotLabel -> U,
Mesh -> None, ColorFunction -> "Rainbow"]}
{Plot3D[W[x, y, Tt] /. First[sol], {x, 0, L}, {y, 0, L},
PlotRange -> All, AxesLabel -> {x, y, ""}, PlotLabel -> W,
Mesh -> None, ColorFunction -> "Rainbow"],
Plot3D[u[x, y, Tt] /. First[sol], {x, 0, L}, {y, 0, L},
PlotRange -> All, AxesLabel -> {x, y, ""}, PlotLabel -> U,
Mesh -> None, ColorFunction -> "Rainbow"]}
It seems to be a direct path into chaos immanent in nonlinear differential equations. The calmed solution is not very good.
It shows the ripples from the calmed solution much more at the ground floor level than atop the great buckle. It provides more similar solutions than the calmed one for all of the four plots. It is not so a great domain than the other ones. The domain is now {-500000,500000} and {-10^7,10^7}. It is not all positive as might be physical, but it is plain in most parts of the defined domain for {t,x,y}.
It first attempted to make the domain smaller. That failed and only proved, time is passing by for the system.
Second I altered the parameters since this seems to gain more insight into the behaviour of the systems under consideration. The did the trick. It resembled, on the other hand, the most important critics from the scientific community for fluid dynamics on the model under consideration. The calming might be due to implicit change in the parameters. That too is possibly still under the regime of chaos in this system.
Nevertheless, it still has potential that chaos is introduced by the methods used to solve the problem and not just the parameters in use. The results presented here are chosen due to physical consideration. As far as I know, this is the first time such results are published for the problem presented here. This is not critics of the methods in NDSolve as offered at present by Wolfram Research.
The computational experiment shows up the fine power of StiffnessSwitching on a very stiff problem with immense singularities of not so point-like character.
