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Topic: Linear system of Hydrodynamic solving by means of PDE toolbox
Replies: 13   Last Post: Jan 16, 2014 10:05 PM

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Bruno Luong

Posts: 9,822
Registered: 7/26/08
Re: Linear system of Hydrodynamic solving by means of PDE toolbox
Posted: Jan 9, 2014 8:22 AM
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"Sergey" wrote in message <la2vkf$rrb$1@newscl01ah.mathworks.com>...
> Dear Matlab users and developers,
>
> I need to solve the following PDE system in 2D space:
> (1) div(grad( Phi )) = 0 |any x, any z;
> (2) Eta_t = Phi_z | any x, z = 0;
> (3) Phi_t = -g*Eta | any x, z = 0.
> where "?_t" - time derivative, "?_z" - z - axis derivative, "div" - divergence operator, "grad" - gradient operator.
> (1) - simple Laplace equation for interior flow potential,
> (2) - surface elevation (Eta) and the potential (Phi) connection,
> (3) - linearized Bernoulli equation.
> It is the linearized equations of hydrodynamics


From (3) and (2) we have

Phi_tt = -g * Phi_z on z = 0; (4)

Consider the (linear) Dirichlet-to-Neumann [DN] map of the laplacian operator:

DN: Phi(z=0) -> Phi_z(z=0) such that div(grad(Phi)) = 0 on { (x,z) : z < 0 }

(4) is second order 1D equation

Phi_tt + g*DN(Phi) = 0 (4bis)

The DN can be solved using PDE toolbox.
The spectrum of DN map then can be used to solve (4bis).

Bruno



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