# Question Regarding Stability of Nonlinear Terms

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 January 2, 2014, 22:52 Question Regarding Stability of Nonlinear Terms #1 Member   Alex Join Date: Jan 2014 Posts: 57 Rep Power: 3 I am new to this forum and new to CFD. I am teaching myself the subject following the structure of "12 Steps to Navier Stokes" (link included) using the finite difference method. The problem with the "12 steps" program is that it omits very important topics such as stability of solutions, uses arbitrary boundary conditions and initial conditions that have no remote connection to the real world, fails to discuss what the actual equations represent. It also uses this garbage shorthand in python, where I am going to be using loops in MATLAB Fortunately I graduated with a BS in Aeronautical Engineering and have some graduate school experience (had to leave for financial reasons sadly), so I have a sufficient fluids background. Regardless, I want to learn CFD and I think a good place to start is the finite difference method. Now onto my question: I am having trouble performing a stability analysis on the nonlinear unsteady convection equation in one and two dimensions (x,y,u,v as variables). I already know how to do a von Neumann stability analysis for 1-D linear systems. From my reading it appears that the VNA cannot be used for nonlinear equations and I have been unable to find a resource that clearly demonstrates a stability analysis procedure for the nonlinear terms. If anyone can help me I would really appreciate it; I'll be able to resume my coding for these equations! The 12 Steps I am following: https://bitbucket.org/cfdpython/cfd-...class/overview

January 3, 2014, 04:29
#2
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 1,642
Rep Power: 23
Quote:
 Originally Posted by H0T_S0UP I am new to this forum and new to CFD. I am teaching myself the subject following the structure of "12 Steps to Navier Stokes" (link included) using the finite difference method. The problem with the "12 steps" program is that it omits very important topics such as stability of solutions, uses arbitrary boundary conditions and initial conditions that have no remote connection to the real world, fails to discuss what the actual equations represent. It also uses this garbage shorthand in python, where I am going to be using loops in MATLAB Fortunately I graduated with a BS in Aeronautical Engineering and have some graduate school experience (had to leave for financial reasons sadly), so I have a sufficient fluids background. Regardless, I want to learn CFD and I think a good place to start is the finite difference method. Now onto my question: I am having trouble performing a stability analysis on the nonlinear unsteady convection equation in one and two dimensions (x,y,u,v as variables). I already know how to do a von Neumann stability analysis for 1-D linear systems. From my reading it appears that the VNA cannot be used for nonlinear equations and I have been unable to find a resource that clearly demonstrates a stability analysis procedure for the nonlinear terms. If anyone can help me I would really appreciate it; I'll be able to resume my coding for these equations! The 12 Steps I am following: https://bitbucket.org/cfdpython/cfd-...class/overview
The VNA (as well as other analysis, such as matrix one) are linear as it is supposed that the error simply summed up the solution, i.e. Un=Uex+err. This implies that the error is simply governed by the same discrete equation you use for the numerical solution.
A non-linear stability analysis is quite more complex, if I remember well, the Hirsch book has some details

 Tags convection, nonlinear, stability analysis

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