Consider the advection-diffusion equation ∂/∂t U + ∂/∂x F(U) = ∂/∂x ( D(U) * ∂/∂x U ). I know that on uniform grids we can use central differences to approximate the advection term as
∂/∂x F(U) |_{i} = ( F(U)|_{i+1} - F(U)|_{i-1} ) / Δx + O(Δx^2),
and the diffusion term as
∂/∂x ( D(U) * ∂/∂x U ) |_{i} = ( D(U)|_{i+1} + D(U)|_{i} ) * ( U|_{i+1} - U|_{i} ) / (2 * Δx^2) - ( D(U)|_{i} + D(U)|_{i-1} ) * ( U|_{i} - U|_{i-1} ) / (2 * Δx^2) + O(Δx^2).
I was wondering if we have second order formulas similar to these for a non-uniform grid, where Δx |_{i} is different than Δx |_{i+1}. I found some works for the advection term, but none for the diffusion one. Is it even possible? Would you provide references? Thank you!
Yes, you can use the Crank-Nicolson scheme and a full second order space discretization.
Filippo Maria Denaro I can find approximations for the advection term on non-uniform grids, but for the diffusion term I struggle to find a second order approximation. Would you have a reference?
Ismael S. Ledoino
for example, you can check the procedure illustrated in the textbook of Ferziger, Peri, Street.
However, you can simply fit a higher order polynomial and evalauate the second derivative.
Ismael Ledoino,
You can use the second-order BEAM-WARMING method to approximate first-order derivatives in time and space. To use a non-uniform mesh, simply apply the Taylor series to non-uniform points and you can derive a formula for this case.
Ismael Ledoino,
You can see the paper "New High-Resolution Central Schemes
for Nonlinear Conservation Laws and Convection–Diffusion Equations" of Alexander Kurganov and Eitan Tadmor. Attached, I send a text with a finite difference approximation of the diffusion term using nonuniform grid.
Dear Geraldo J B Dos Santos
the question in the post is not mine, I am definitely aware of many numerical discretizations to get a second order (and higher) time and space accuracy on non uniform grids.
However, many thanks for your reference.
Dear Filippo Maria Denaro,
I know. I have already corrected the recipient. It was the mistake.
Thank you Geraldo J B Dos Santos , for your contribution. I have myself generated a similar formula for non-uniform grids. However, by keeping track of the remainders I noticed my formula did not have a second order truncation error, which is why I posed the question. Interestingly, numerical methods showed the convergence was indeed second order and that confused me. When Filippo Maria Denaro gave me the reference above, I finally understood that even though the truncation error may be first order, the convergence is assymptotically second order. That clarified this for me, thank you all!
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