I have some nonlinear PDEs that I wish to solve numerically. My initial stab at the solution seems to be very naive. I discretised the PDE using finite differences, and this leaves me with a set of nonlinear algebraic equations at each time step. To my simplistic mind, I can solve these using the Newton-Raphson method.
I tried this method and I can't get the solution to converge for some reason. Was my idea wrong from the outset?
This is not a good plan of attack, as you have discovered. Nonlinear PDEs are a challenge but much has already been learned. CFD is an excellent example. There are many approaches to solving the Navier-Stokes Equation (coupled with the Continuity Equation plus an approximation to handle turbulence). Which method works "best" depends on the characteristics of the problem. All this is to say that what approach might work best for your problem requires more information. I can probably come up with something, having written thousands of programs and millions of lines of code. I have probably written a book on the subject too. For example, Numerical Calculus is free tomorrow https://www.amazon.com/dp/B07BS1DN1S
Hi Mat! I hope that all is well.
I use NR to solve sets of discretised nonlinear equations all the time. The first type of problem that I used the method on were nonsimilar boundary layer equations where I used the Keller-box method. More recently I've solved a set of equations for what are, in essence, Fourier coefficients. This system is very highly nonlinear.
In both cases (and others) the Jacobian matrix is computed within the code rather than specified by the programmer.
It may be necessary for some applications to solve each NR iteration using a block tridiagonal solver.
I don't mess around with Matlab for this. Good old-fashioned Fortran works very well for me.
I will say that, once you've cracked it you'll be dancing. So yes, persevere!
...I forgot to ask, what is the nature of the ODE. Are you trying to solve an unsteady 1D problem (e.g. a nonlinear version of Fourier's equation), an unsteady 2D problem (nonlinear 2D Fourier), or a system of coupled equations?
Andrew makes a very good point: stay away from MATLAB. For a good laugh on how pathetically slow python is, watch this hysterical video. https://www.youtube.com/watch?v=c33AZBnRHks
I tried the NR idea with a nonlinear ODE and then the viscous Burgers equation. The technique worked well with the first one, and worked very well up to a point with the VBE and then the solution defaulted to being zero.
Initially I was solving a 1D system of equations that included conservation of mass, momentum, and energy. The conservation of mass is hyperbolic whereas the other two are parabolic, and this causes issues. I then went to a linear system that represented my nonlinear system. I thought the NR method would work very well, but it didn't even converge.
I don't like Python because the syntax is absolutely horrible, it reminds me of C++ it's so bad. The fact that you don't close your loops with an end I think could cause serious issues in programming.
Matlab has nice syntax, and you can immediately plot things without having to include anything else.
The equations I want to solve are (1)-(4), which are very nonlinear. I then looked at a linear version of the equations (19)-(21), but I still had issues.
I compute the Jacobian at runtime, and as I said before it seems to work very well for a simple ODE and still looks okay, likewise with the viscous burgers equation(up to a point). It seems to fail completely for systems of mixed type equations though.
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