# Finite Difference Method, Leapfrog (2,4) CFL Condition

• wrechtin
In summary, Will is looking for the CFL condition for the fourth-order leapfrog scheme and is finding a published value of 2, which does not match his calculations. He is seeking a von Neumann or Fourier stability analysis of the leapfrog (2,4) scheme for comparison. He clarifies that he is not referring to numerical experiments, but rather the theoretical CFL condition and von Neumann stability analysis. He wants to determine the theoretical CFL condition in order to properly design his experiments and ensure a solid understanding of the topic.
wrechtin
Hi.

I'm trying to determine the CFL condition for the fourth-order leapfrog scheme. I'm finding 2 as what's published, which does not match what I'm getting.

Does anyone know where I can find a von Neumann (or Fourier) stability analysis of the leapfrog (2,4) scheme (so I can compare my work) and/or a reputable book dedicated towards von Neumann (or Fourier) stability analysis?

Will

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When you say "your calculations", do you mean numerical experiments with a finite number of mesh points and some specific boundary conditions? Or do you mean your own attempt at a theoretical stability analysis?

If you mean numerical experiments, I would not be surprised that 2 is the limit for an infinitely large mesh. Try mesh sizes of n, 2n, 4n, 8n... for some reasonable value of n, and see what happens.

Hi AlephZero.

Thank you for the reply. No, I'm not referring to numerical experiments. I'm referring to the theoretical CFL condition and to a very basic application of von Neumann stability analysis where u^m_n = (g^m)(exp(i*xi_n*h)) and the scheme is u^(m+1)_n = u^(m-1)_n + lambda((4/3)(u^m_(n+1) - u^m_(n-1)) - (1/6)(u^m_(n+2) - u^m_(n-2)). I would prefer to determine the theoretical CFL condition, so I know what step sizes to gather my data in for preliminary confirmation of fitting the predicted with the observed and then subsequent experiments. I could just accept what's already been published, but I want to make sure I have a solid understanding of what's in front of me before I progress.

Are you implying the published CFL condition is referring to actual numerical experiments or is it theoretical?

Will

## 1. What is the Finite Difference Method?

The Finite Difference Method is a numerical method used to approximate solutions to differential equations. It involves dividing a continuous domain into a discrete grid and using finite difference approximations to calculate the values of the unknown function at each grid point.

## 2. How does the Leapfrog (2,4) CFL Condition work?

The Leapfrog (2,4) CFL Condition is a stability criterion used in the finite difference method for solving time-dependent problems. It ensures that the time step used in the solution is small enough to accurately capture the dynamics of the system. It requires that the time step is less than half of the spatial grid size divided by the maximum wave speed in the system.

## 3. What is the significance of the CFL number in the Leapfrog (2,4) CFL Condition?

The CFL number, also known as the Courant-Friedrichs-Lewy number, is a dimensionless parameter used to determine the stability of a numerical solution to a time-dependent problem. In the Leapfrog (2,4) CFL Condition, the CFL number must be less than 1 for the solution to be stable.

## 4. Can the Leapfrog (2,4) CFL Condition be used for all types of differential equations?

No, the Leapfrog (2,4) CFL Condition is specifically designed for hyperbolic partial differential equations, which describe wave-like phenomena. It may not be applicable to other types of differential equations, such as parabolic or elliptic equations.

## 5. How does the Leapfrog (2,4) CFL Condition compare to other stability criteria in the Finite Difference Method?

The Leapfrog (2,4) CFL Condition is known for being more restrictive compared to other stability criteria, such as the Forward Euler method. This means that it may require smaller time steps to achieve stability, but it also leads to more accurate solutions. Other stability criteria may allow for larger time steps, but the solutions may be less accurate.

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