reflective boundary conditions pdepe conservation of mass: please HELP!
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From: Kyle on 14 Jul 2010 09:42 Hello, I am having trouble with diffusion with drift and the conservation of mass in pdepe. The model I am trying to simulate is described as follows: Within a container is a fixed number of particles in solution that is subject to diffusion and an external force. The external force is analogous to a uniform electric field that influences each particle in the same direction and magnitude no matter its position within the container. The variable of interest will be the concentration of particles at a given position and time. I want my simulation to show the steady state behavior of such a system. The conservation of mass should hold as no matter will enter or leave the container. I would expect that the mass will collect along the side of the container perpendicular to the applied force and form a gradient in the opposite direction of the force, due to the equilibrium between diffusion and the flow created by the force. To model this as simply as possible the container is reduced to a single dimension, an interval of fixed length [0,L]. The concentration, u, at any given point and time will satisfy this equation, u_t = -D*u_xx + V*u_x where u_t is the partial derivative w.r.t. time and u_xx is the second partial derivative w.r.t. its position x, etc. There can be any arbitrary initial condition u0 and boundary conditions... This is where I begin to get confused... I am uncertain about what boundary conditions I should use and how to enter these into MATLAB's pdepe. I believe I am supposed to arrange some kind of reflective boundary conditions, but I am uncertain how to apply this in practice. I do not want to fix the concentrations at the edges, because the concentrations should be allowed to change there. I first thought I should arrange a no flux condition and set the Neumann conditions to zero, but this is not the answer, my solutions never reach steady state. However, when I remove the "drift" term, i.e. set V=0, diffusion reaches a steady state and conservation of mass appears to hold. I then learned about reflective boundary conditions, but do not know how to implement them or if this is even correct. If anyone has an answer, could you please give me an explanation and the how to implement these boundary conditions in the pde format of "pl=?, ql=?, pr=?, qr=?"? Thanks, KWK |