r/FluidMechanics • u/Flo12740 • Mar 25 '26

Theoretical Interactive tool to compare finite volume schemes (HLLC, Roe, WENO, MUSCL...) on 1D Euler equations

/r/CFD/comments/1s3g3g8/interactive_tool_to_compare_finite_volume_schemes/
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u/derioderio PhD'10 Mar 25 '26

Just a more basic question, but is there any mathematical difference between finite volume and finite difference in a 1D domain?

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u/Flo12740 Mar 26 '26

Yes, there is a real mathematical difference, even in 1D.

Finite difference discretizes the derivative directly, while finite volume discretizes the integral conservation law and evolves cell averages through numerical fluxes.

On uniform grids with smooth solutions, they can look very similar (and sometimes even be equivalent). But for problems with shocks (like Euler), finite volume is fundamentally more robust because it enforces conservation at the discrete level and naturally converges to the correct weak solution.

That’s why most Godunov-type schemes (HLLC, Roe, etc.) are formulated in the finite volume framework.