1. The Courant-Friedrichs-Lewy Condition¶

Comments on the 12 Steps to Navier Stokes
Stability and Convergence, Modified Differential Equation and Truncation Error
Summary of Stability
- Implicit and Explicit Schemes
Physical Interpretation of the CFL Condition

1.1. Comments on the 12 Steps to Navier Stokes ¶

We have really used the simplest possible methods for the 12 Steps to Navier-Stokes
These methods are unsuitable for practical purposes, because of diffusion e.g. the use of 1st order upwinding (BD in space) for the convection term in the momentum equation for example.
We must now develop new methods that are higher order methods.

1.2. Stability and Convergence, Modified Differential Equation and Truncation Error ¶

For 1D convection, CD in space with FD in time produced a Modified Differential Equation, which was not a convection equation, but a convection-diffusion equation with a negative diffusion coefficient (helping to explain why the scheme was unstable).

\[\bar{u}_t + c \bar{u}_x = -{{\Delta t} \over 2} c^2 \bar{u}_{xx} + O({\Delta t^2}, \Delta x^2)\]

Negative diffusion coefficient represents an explosion (unstable)

In contrast, 1st order upwind for 1D convection:

\[\bar{u}_t + c \bar{u}_x = {{c \Delta x} \over 2} (1+{{c \Delta t} \over {\Delta x}}) \bar{u}_{xx}\]

Contained numerical diffusion proportional to \(\Delta x\) (first order) which is why the diffusion is excessive

Stability condition \(c \gt 0\) and \(\sigma = {{c \Delta t} \over \Delta x}\) (CFL Condition)

Then we performed von Neumann analysis for stability

1.3. Summary of Stability ¶

Schemes can have:

Conditional Stability
Unconditional Stability
Unconditional Instability

1.3.1. Implicit and Explicit Schemes ¶

Implicit schemes are generally unconditionally stable. Implicit methods have no such restriction on time step, but require more computation per timestep (you must solve a linear system of equations)

Explicit schemes lead at best to conditional stability. Conditional stability puts a limit on the timestep - we cannot progress too rapidly in time with the numerical scheme - as it may nnot be able to “trasmit” the information in the solution

Convection: The limit on the time step can be a severe requirement, especially for convection dominated equations (c is large compared to other effects). For most explicit schemes:

\[{{c \Delta t} \over {\Delta x}} \le 1\]

Diffusion: For diffusion dominated equations, the restriction is less severe, i.e.

\[{{\nu \Delta t} / {\Delta x^2}} \le {1 \over 2}\]

1.4. Physical Interpretation of the CFL Condition ¶

1st order upwind CFL condition (for most explicit schemes):

\[{{c \Delta t} \over {\Delta x}} \le 1\]

Physical Interpretation: The distance travelled by the solution in one timestep \(c \Delta t\) must be less than the distance between two mesh points \(\Delta x\)

Hyperbolic PDEs (such as the one for linear convection) have characteristic lines along which the solution travels
For upwinding (backward differencing in space) the domain of dependence at n is from i-1 to i. This is suitable for a positive wave speed c
For forward differencing the domain of dependence at n is from i to i+1. This is suitable for a negative wave speed c
The solution at the next timestep must be able to include all the physical information that influences the solution from the previous timestep
The CFL condition \(\sigma \lt 1\) ensures that the domain of dependence of the governing equation is entirely contained in the domain of dependence of the numerical scheme
Can extend this to more complex cases where deriving the stability condition is more difficult for more complex numerical schemes.
Also demonstrates why backward differencing is unstable for a negative wave speed, i.e. if the wave move from right to left, the solution should depend on points i and i+1 not i and i-1

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