[aerosayan]

I was testing with calculating local dt to satisfy CFL condition for a 1d non-linear convection solver.



I used the formula dt = cfl*dx/un[i]

When CFL was 1.0, the wave didn't have have any numerical diffusion, but for CFL 0.9 the numerical diffusion started to occur.

Why? For CFL 0.9, the code should be more stable, right?


I still don't 100% understand: How does numerical diffusion happen?


Julia code:

using Plots;

n      = 1000; # no. of nodes in the domain
c      = +1.0; # linear velocity of the wave

xstart = +0.0;   # start of domain
xend   = +100.0; # end of domain

dt     = +99999999;             # timestep (*needs* to be computed later to satisfy cfl condition)
dx     = (xend-xstart)/(n-1.0); # spacestep
cfl    = 1.0;                   # courant-freidrichs-lewy condition

icstart = 25;  # initial condition start index
icend   = 100; # initial condition end index

x = collect(LinRange(xstart,xend,n)); # domain or mesh of n nodes
u = collect(LinRange(1.0,1.0,n));     # solution of the linear wave equation
u[icstart:icend] .= 2.0;              # set initial condition, a hat function
u0 = copy(u);                         # initial solution

for t in 1:2^8; # timestep iterations
    un = copy(u);
    for i in 2:n
        dt   = cfl*dx/un[i];
        u[i] = un[i] - un[i]*dt/dx*(un[i]-un[i-1]);
    end
end

display(plot(x,[u0,u]));

[FMDenaro]

Sorry but what about your background in CFD?

This is one of the fundamental topics in any CFD course! Just start considering the analysis in the physical space by adopting the modified differential equation technique. Have a look to the evaluated expression that multiplies the second derivative in space.
Then you can also see the same issu in the wavenumber space.


That has no need of any code to be seen. In your code I see that you compute a different dt value for different location i and that is wrong!
I also suggest to use the conservative form, not the quasi-linear form of the Burgers equation.

[aerosayan]

Thanks for your help! I'm self-taught, with some gaps in knowledge.

[FMDenaro]

Originally Posted by aerosayan

Thanks for your help! I'm self-taught, with some gaps in knowledge.

Be careful, you need to learn carefully the basics...

Your code has the first order upwind with backward formula but you have to consider that the Burgers equation can have initial negative data and your code will not work.


PS: I strongly suggest to check what happen using a Heaviside initial condition. Compare the velocity of your shock wave with the theoretical value.

[sbaffini]

For the linear, pure convection case in 1D with uniform grid, first order upwind and first order explicit Euler (I don't know for other cases) it can be shown analytically that CFL = 1 leads to exactly 0 numerical error (diffusion included) because ALL the spurious terms in the MDE become 0.

Of course, if you think about it, it is as much obvious as useless in real cases. Yet, it tells you something important about numerical convection that lagrangian schemes explicitly try to leverage.

[aerosayan]

Originally Posted by sbaffini

For the linear, pure convection case in 1D with uniform grid, first order upwind and first order explicit Euler (I don't know for other cases) it can be shown analytically that CFL = 1 leads to exactly 0 numerical error (diffusion included) because ALL the spurious terms in the MDE become 0.

Didn't know that. Thanks!

[arjun]

if you can find this book

https://link.springer.com/book/10.10...-94-007-4038-9


There is a detailed discussion about what you have asked. What you asked is not a simple question.


A very very short and crude answer to your question is that at cfl of 0.9 the wave moves somewhere in between the cells and now the face values are not well constructed (for any reason, mostly it is limiting that would do this). Thus you end up with all sorts of issues, this diffusion is one of them.