12 votes
Accepted

Understanding the Courant–Friedrichs–Lewy condition

I have two extra points I would like to add to Wolfgang's answer. A formulation of the CFL condition that I find more useful than the classic formula is this: A necessary condition for the ...
David Wells's user avatar
9 votes

Understanding the Courant–Friedrichs–Lewy condition

You are correct: If you satisfy the CFL condition, then all that guarantees is that your scheme is stable, i.e., the numerical solution does not go to infinity. But the CFL condition says nothing ...
Wolfgang Bangerth's user avatar
5 votes

Time step relationship with number of elements or material properties

It sounds as if you're running a time-dependent linear elasticity simulation, right? Most likely, you're running an "Explicit" time-stepping scheme, which means that all of your information at time $...
Tyler Olsen's user avatar
  • 1,522
4 votes

Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?

CFL is a necessary requirement for stability in transport-dominated systems, but in practice, is an intuitive way to interpret linear stability. A 1D advection-diffusion (or somewhat upwinded ...
Jed Brown's user avatar
  • 25.7k
4 votes

Proof of CFL condition for RKDG scheme

As far is I know, a proof for the general case has not been presented so far. The most detailed analysis has probably been performed by Krivodonova & Qin 2013 in this work. Quoting from the ...
bigge's user avatar
  • 293
4 votes
Accepted

CFL condition in polar coordinates

I was going to write a comment, but the equation seems to view better in answers.. I assume Von Neumann analysis is the proper approach to derive this equation, but a coordinate transformation from ...
Charles's user avatar
  • 619
4 votes

Time step relationship with number of elements or material properties

The most commonly used explicit ODE solver in structural analysis is the central difference method. Because it is explicit, the solution becomes unstable if the time step is larger than a so-called ...
Bill Greene's user avatar
  • 6,064
3 votes
Accepted

CFL condition in Stokes equation

Essentially, the time dependent Stokes equation looks like the heat equation: $$ \frac{\partial u}{\partial t} - \nu\Delta u = f-\nabla p, $$ plus the incompressibility condition $\nabla \cdot u=0$ ...
Wolfgang Bangerth's user avatar
2 votes

For implicit schemes, is there any general result that says numerical diffusion increases with smaller timesteps (for CFL<1) as in explicit schemes?

Short Answer: There is no general result that would hold for all implicit schemes. The reason is that how your method behaves with respect to numerical diffusion depends on the specific combination of ...
Daniel's user avatar
  • 1,273
2 votes

Finite elements with CFL condition - How to obtain correct order of convergence

This is one of those scenarios where assessing the convergence order of your scheme is difficult, because, as you explained, your time-step is "linked" to your spatial discretization. What ...
BlaB's user avatar
  • 1,157
2 votes
Accepted

Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?

Yes. That's all there is to the stability condition. Taking the material properties - shear modulus ($\mu$), bulk modulus ($\kappa$) and density ($\rho$) - into account, the global critical time step ...
Chenna K's user avatar
  • 934
2 votes
Accepted

CFL equation for non-linear equation

For the simplest case of linear advection, applying von Neumann stability analysis gives the necessary restriction on the time step size, known as the CFL condition: $$ \frac{c\Delta t}{\Delta x}\le ...
fraktaali's user avatar
2 votes

Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?

The CFL condition refers strictly to purely hyperbolic problems where there is a well-defined domain of dependence for the analytical solution. It state that a necessary condition for stability is ...
Philip Roe's user avatar
  • 1,154
1 vote

Stability Criterion for this Explicit Scheme

It is done in the standard way through von Neumann stability analysis. The CFL condition on the pseudo timestep is straight forward. The restrictions due to the viscosity and other terms are a little ...
Abhilash Reddy M's user avatar
1 vote

CFL condition and Lax-Friedrich numerical flux

I believe to some extent there is confusion due to the name of the scheme. The "classic" Lax-Friedrichs Scheme is model-agnostic in the sense that no information on the flux function $f(u)$ ...
Dan Doe's user avatar
  • 1,083

Only top scored, non community-wiki answers of a minimum length are eligible