The discussion centers on the divergence observed in a forward time-centered space finite difference scheme when increasing the final run time (##tf##). The participants identify that the stability of the numerical method is influenced by the Courant-Friedrichs-Lewy (CFL) condition, which states that ##\frac{\Delta t}{\Delta z} < C## for stability. A von Neumann stability analysis is recommended to derive necessary conditions for stability, particularly for nonlinear equations. The conversation also touches on the importance of initial conditions and the potential for linearizing the problem to facilitate analysis.
PREREQUISITESResearchers and practitioners in computational fluid dynamics, numerical analysis, and applied mathematics, particularly those working with finite difference methods and stability analysis of PDEs.
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.