The discussion revolves around the behavior of a finite difference scheme used to solve a nonlinear partial differential equation (PDE) and the observed divergence when increasing the final run time. Participants explore the implications of time step size and stability criteria in numerical methods, particularly in relation to the CFL condition and the effects of spatial domain size.
Participants express differing views on the application of stability criteria, particularly regarding the use of von Neumann analysis for nonlinear equations. There is no consensus on the best approach to ensure stability in the finite difference scheme, and multiple competing views remain regarding the reformulation of the PDE and its implications for numerical stability.
Participants highlight limitations in their approaches, including the dependence on specific assumptions about the PDE and the potential for inaccuracies in the reformulated equations. The discussion also reflects uncertainty about the effects of initial conditions on the numerical results.
This discussion may be useful for individuals interested in numerical methods for solving partial differential equations, particularly those exploring finite difference techniques and stability analysis in nonlinear contexts.
You did it so you could always have a grid point exactly at the moving boundary, making it convenient to apply any boundary condition there. Also, you always have the same number of grid points, and they are uniformly distributed between the boundaries. You've taken the complexity out of applying the moving boundary condition and transferred it to the differential equation where it can be handled much more easily.joshmccraney said:Hey Chet, I have a quick question. When transforming from ##Z## to ##z## I did the following: ##Z=L(tf)z##, but now I'm not sure why I used the last value of ##L##. Should ##L## be evaluated at the final time? Could you explain your reasoning?
It depends. What you would do would be to only discretize the differential equations in the spatial direction, but not in the time direction. So then you would have a coupled set of non-linear first order ODEs in time for the y values at the M grid points (plus an additional first order ODE for L2). This approach is called the Method of Lines. To solve these equations implicitly, you would typically need a stiff ODE equation solver subroutine. The problem would be done in FORTRAN, and you would be using a commercially available stiff package like the ones available on IMSL or online (e.g., DASSL). Or, if Matlab has a stiff equation solver, you could use that.joshmccraney said:Hey Chet, sorry to bother you again about this problem, but how much more effort would need to be put into the finite difference scheme to go implicit? I've never done this before and am curious about it.