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I'm using the finite difference method to solve both the Reynolds Equation and the Poisson equation in order to calculate the air pressure of a porous air bearing. The Reynolds equation describes the flow in the channel between the rotor and stator, while the Poisson equation the flow through a porous material describes. However, the pressure in the channel, calculated by the Reynolds equation, and the pressure in the porous material at r = ri (inner radius of the porous cilinder) must be equal to each other in order to ensure continuity of the pressure at r = ri. Because both equations are solved iteratively, the loop is only stopped from the moment that both pressures are equal to each other at r = ri.

During this iterative process, the pressure distribution in the channel calculated by the Reynolds equation at step i, is used in step i+1 as a boundary conditions for calculating the pressure distribution within the porous material from the Poisson equation. This result of the pressure distribution within the porous material, is then used to solve the Reynolds equation again. This process repeats until the pressure difference at r = ri, from both equations is lower than a certain percentage. However, the convergence rate of this method is very slowly.

Does anyone know how I can increase the convergence speed? I was thinking about the Newton-Raphson method or the Regula Falsi method. However, I don't really know how to do this for a 2D problem (finite difference grid for the pressure distribution between the rotor and stator is 2D).

Any help or example is welcome!!

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# How to increase the convergence speed?

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