
Re: Linear system of Hydrodynamic solving by means of PDE toolbox
Posted:
Jan 9, 2014 8:22 AM


"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) DirichlettoNeumann [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

