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August 29, 2017, 21:16 |
Discretization of the diffusion term
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#1 |
New Member
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For incompressible NS equation in the finite volume discretization on a staggered grid, I have encountered a doubt.
We can put the diffusive term together with the advection term by factoring out gradient, for instance for x-direction for one term as below: I am fine with the way to discretize the second term in staggered grid but for the first term in the code that I see someone has discretized the first term as Why is this correct? Even if it is the central difference, shouldn't it also include ? |
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August 29, 2017, 21:29 |
Diffusion
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#2 |
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Selig
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The finite difference discretization below ensures proper conservation.
The advection term of the form is not conservative and can lead to the numerical solution undershooting. As a side note, central differencing like that of the advection term can lead spurious oscillations. QUICK scheme or a stable advection scheme (ENO, WENO) can remedy such a problem. |
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August 29, 2017, 21:36 |
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#3 |
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So you mean all is good with the conservative form of discretization? And no oscillations occur?
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August 29, 2017, 22:02 |
Conservative schemes
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#4 |
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Selig
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Yes, you will want your scheme to be conservative. In terms of your second question, spurious oscillations can occur regardless if your scheme is conservative or not. A simple 1D example would be looking at the convection-diffusion equation and adjust the peclet number. You will find that upwind differencing will perform better than central differencing.
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August 30, 2017, 03:59 |
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#5 |
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Filippo Maria Denaro
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I agree, consider the Burgers equation in divergence and non-divergence (quasi-linear) form:
du/dt + d(u^2/2)/dx=d(mu*du/dx)/dx (divergence form) du/dt + u*du/dx=mu*d2u/dx^2 (quasi linear) No difference exists for the staggered or non staggered grid since the only involved variable is u. Discretize with second order central difference a see what happens. The appearence of u(i,j) in the convective term: d(u^2/2)/dx -> (u(i+1/2)^2-u(i-1/2)^2)/2/h du/dx -> (u(i+1/2)-u(i-1/2))/h Now you have two chance: 1) linear interpolation for u on the half-nodes 2) linear interpolation for u^2 on the half nodes The first choice is congruent to a linear reconstruction of the flux function, the second one introduces aliasing in the terms. |
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August 30, 2017, 10:41 |
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#6 |
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Filippo Maria Denaro
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numerical oscillations are due to the lack of monotonicity of the scheme... Godunov theores states that monotonic linear scheme can be only first order accurate. Have also a look to the examples in the book of Peric & Ferziger
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discretization scheme, navier stokes |
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