# Solve a Saint-Venant PDE with NDSolve?

Posted 7 months ago
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 Hi,I try to simulate the equalization/distribution of a fluid within a confined volume. I selected the shallow water equations (Saint-Venant equations) and removed the slope and friction part of the equations. The two equations I use are:eqn1 = eqn2 = With: w - constant width of the channel h - height of the water in the channel u - flow velocity t - time x - longitudinal axis along the channel g - gravity constant As boundary condition I use: boundaryCondition = {u[-1, t] == 0, u[1, t] == 0} which shall be equivalent to "no flow at x=+-1". My initial condition is no flow at t=0 anywhere and some hat like distribution of the fluid height between x=-1 ... 1: initialCondition = {h[x, 0] == 3 + Cos[x*\[Pi]/2]^2, u[x, 0] == 0}; Cos[x*\[Pi]/2]^2:Before solving I assign the remaining constants: constAssignments = {w -> 1, g -> 9.81}; After that I call NDSolve with the following parameters which unfortunately lead to a warning/error: sol = NDSolve[{eqn1 /. constAssignments, eqn2 /. constAssignments, initialCondition, boundaryCondition}, {u, h}, {x, -1, 1}, {t, 0, 10}] NDSolve::ndsz: At t == 1.9560944493733925, step size is effectively zero; singularity or stiff system suspected. NDSolve::eerr: Warning: scaled local spatial error estimate of 4.424896536902471*^7 at t = 1.9560944493733925 in the direction of independent variable x is much greater than the prescribed error tolerance. Grid spacing with 25 points may be too large to achieve the desired accuracy or precision. A singularity may have formed or a smaller grid spacing can be specified using the MaxStepSize or MinPoints method options. A plot of the results with Plot3D[h[x, t] /. sol, {x, -1, 1}, {t, 0, 1.5}]` reveals that it looks pretty good for the first tenth of a second but then the result gets disturbed.Do you have any suggestion how to solve this issue? I also wonder why DSolve is unable to solve this problem symbolically since the solution should be quite simple. I have attached the notebook file with the problem and some further explanations. Thank you! Attachments:
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Posted 7 months ago
 Maybe this helps.Maple 2018 can't find symbolic solution,only a general.See attached file and ComparisonRegards,MI Attachments: