Questions tagged [cfl]
Courant-Friedrichs-Lewy number is a condition of stability of numerical discretization scheme for a time dependent PDE
26
questions
1
vote
0
answers
96
views
Stability for the 2d diffusion equation
I'm reading this set of notes on numerical stability of the diffusion equation. I can't seem to derive equation (7.17) from the previous equation. I've tried using $\sin^2(\theta)=\frac{1-\cos(2\theta)...
- 668
2
votes
3
answers
152
views
Should reaction be taken into account in the CFL condition when solving advection-diffusion-reaction equations numerically?
I'm dealing with the numerical resolution of advection-diffusion-reaction equations.
I've encountered references on CFL conditions for conservation laws as well as advection-diffusion equations.
I'm ...
- 61
1
vote
1
answer
1k
views
How to solve heat equation in spherical coordinates with finite differences?
I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only.
...
- 111
3
votes
1
answer
476
views
For implicit schemes, is there any general result that says numerical diffusion increases with smaller timesteps (for CFL<1) as in explicit schemes?
For the first-order explicit upwind scheme, it can be easily shown that, if one keeps the same grid size and progressively decreases the time step below the max allowed one (i.e. below CFL~1) the ...
- 435
0
votes
1
answer
372
views
Finite elements with CFL condition - How to obtain correct order of convergence
I have discretized a PDE with continuous finite element method in spatial variable and with implicit Euler or Crank-Nicolson in temporal variable. In both cases, I have error estimates in $L_2$ norm ...
- 145
0
votes
1
answer
195
views
Technique to find the CFL condition using the Galerkin method in space and finite-difference in time?
I am using the Galerkin method (Discontinuous to be precise) to discretize in space the scalar linear wave equation and the explicit second order centered finite difference scheme to discretize in ...
- 155
1
vote
0
answers
81
views
Are linear, CTCS codes always stable?
I would like to solve some equations which basically look like this
$$\frac{\partial u}{\partial x}=F\left(v,\frac{\partial v}{\partial y},\frac{\partial^2 v}{\partial y^2}\right),$$
$$\frac{\partial ...
- 219
2
votes
0
answers
47
views
Verification on pressure predictor method for CFD code
I have developed a python code for a lid-drive cavity model. However, my results are not converging. The algorithm of my code looks like this:
Euler Momentum Equation looks like this:
$$\frac{u^{n+1}...
- 123
2
votes
2
answers
1k
views
CFL equation for non-linear equation
I am trying to solve numerically (obviously) inviscid Burgers' equation with the finite difference method. The equation is the following:
$$ \displaystyle \partial_t u + u \, \partial_x u = 0 $$
...
user30503
1
vote
0
answers
267
views
Definition of CFL number in Arbitrary Lagrangian-Eulerian framework
In an Eulerian frame of reference, the CFL number is defined as $$\sigma=\frac{u \Delta t}{\Delta x}$$
with $u$ the magnitude of the fluid velocity. A restriction such as $\sigma<1$ for time ...
- 1,055
2
votes
1
answer
2k
views
CFL condition in Stokes equation
Does the CFL condition play any role in a pure Stokes flow, i.e. convective term is neglibile, or vanishing? If not, what is the "equivalent" condition for stability? I have read something about the ...
- 45
1
vote
0
answers
320
views
CFL condition of source term
Consider one dimensional hyperbolic pde
$$u_t+f'(u)u_x=0$$
For the above problem ,CFL condition is $\Delta t\leq \dfrac{\Delta x}{|f'(u)|}.$
But if we include the source term $,S(u,t),$ which ...
- 33
0
votes
2
answers
1k
views
Time step relationship with number of elements or material properties
When looking at the output file of my solver, I have been told that the time-step taken by the solver depends on parameters like the total number of elements and their relative size in my geometry, or ...
- 161
9
votes
1
answer
817
views
CFL condition in polar coordinates
In this question, I suggested that the Couran-Friedrichs-Lewy (CFL) condition for the wave equation in polar coordinates reads
$$C = 2c\frac{\Delta t}{\Delta r \Delta \phi} \leq C_\max \enspace ,$$
...
- 8,370
12
votes
2
answers
12k
views
Understanding the Courant–Friedrichs–Lewy condition
I understand these equations in particular can be solved easily without use of computational methods. Although right now I am concerned with trying to solve these equations using numerical integration ...
- 578
3
votes
1
answer
283
views
Stability Criterion for this Explicit Scheme
I am solving an unsteady flow using the dual-time Navier-Stokes equation in which I write my momentum equation as:
$$\frac{\partial u}{\partial \tau} + \frac{\partial u}{\partial t} + \frac{\partial u^...
- 414
5
votes
1
answer
370
views
Proof of CFL condition for RKDG scheme
The cfl condition for linear advection equation
$$
u_t + a u_x = 0
$$
using a DG method of degree $k$ polynomials, upwind flux and an RK scheme of $k+1$ stage/accuracy is stated to be $\frac{1}{2k+1}$...
- 2,993
0
votes
1
answer
814
views
Find cfl condition
We have the advection equation $u_t+a u_x=0, a>0, 0<t<T_f, x \in \mathbb{R}$ with initial condition $u(0,x)=u_0(x)$.
Suppose that we have the following sheme:
I want to find the CFL ...
- 149
0
votes
1
answer
105
views
How do we find the condition?
Suppose that we are given a numerical scheme.
In order to find the CFL condition , we set $U_j^n= \lambda ^ne^{ik x_j}$ and put it into the numerical scheme.
I have shown that the given method is ...
- 149
5
votes
0
answers
2k
views
CFL Condition and Convection Diffusion Equation in 2D
I am solving the convection-diffusion equation in 2D using Finite Differences with the $\theta$ scheme. The velocity of the fluid and the diffusion coefficient is low in my case (in the range of $10^{-...
- 159
4
votes
1
answer
1k
views
CFL condition for variable coefficients
I understand that the constant in the Courant-Friedrichs-Lewy condition is defined as $\mathrm{CFL} = \frac{u \Delta t}{\Delta x}$, where $u$ is the principal coefficient. I came across this post:
...
- 169
4
votes
2
answers
697
views
CFL for high order finite difference
Time-explicit high order finite element/DG methods for the advection equation tend to have a timestep restriction which looks like
$$dt < C h/p^2$$
where $h$ is mesh spacing and $p$ is polynomial ...
- 3,112
8
votes
3
answers
238
views
Stability criterion for waves in anisotropic solids
The equations of motion for an elastic solid are given by
$$\begin{align}
&\nabla \cdot \boldsymbol{\sigma} + \mathbf{f} = \rho \ddot{\mathbf{u}}\\
&\boldsymbol{\sigma} = \mathbb{C}\...
- 8,370
2
votes
1
answer
2k
views
CFL condition and Lax-Friedrich numerical flux
I got confused when trying to implement a scheme using Lax-Friedrichs numerical flux for a system of equations in 1D.
According to my notes Lax-Friedrichs numerical flux is
$$f_{LF}(u_l,u_r) = \frac{...
- 1,019
3
votes
1
answer
1k
views
Von Neumann stability analysis in 3d
I need to get a stability criterion for the numerical scheme for equation
$$\frac{\partial u}{\partial t}-\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}-\frac{\partial^2 u}{\...
- 491
4
votes
1
answer
2k
views
Courant Friedrichs Lewy condition - how to get it?
I am interested, how can we get CFL condition for every type of PDE? It's known that for 1st order linear equation
$$\frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0 $$
CFL is get from ...
- 491