Discretization of the diffusion term

mseka · August 29, 2017, 22:16

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:
$\frac{\partial }{\partial x}\left ( u^2-2 \mu \frac{\partial u}{\partial x} \right )$

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
$\frac{u^2_{i+1,j}-u^2_{i-1,j}}{2\Delta x }$

Why is this correct? Even if it is the central difference, shouldn't it also include $u_{ij}$ ?

selig5576 · August 29, 2017, 22:29

The finite difference discretization below ensures proper conservation.
$\frac{u_{i+1}^{2} - u_{i-1}^{2}}{2 \Delta x}$

The advection term of the form
$u_{i} \frac{u_{i+1} - u_{i-1}}{2 \Delta x}$

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.

mseka · August 29, 2017, 22:36

So you mean all is good with the conservative form of discretization? And no oscillations occur?

selig5576 · August 29, 2017, 23:02

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.

FMDenaro · August 30, 2017, 04:59

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.

FMDenaro · August 30, 2017, 11:41

Quote:

Originally Posted by mseka

So you mean all is good with the conservative form of discretization? And no oscillations occur?

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

August 29, 2017, 22:16	Discretization of the diffusion term	#1
mseka New Member Join Date: Jul 2017 Posts: 6 Rep Power: 9	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: $\frac{\partial }{\partial x}\left ( u^2-2 \mu \frac{\partial u}{\partial x} \right )$ 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 $\frac{u^2_{i+1,j}-u^2_{i-1,j}}{2\Delta x }$ Why is this correct? Even if it is the central difference, shouldn't it also include $u_{ij}$ ?

August 29, 2017, 22:29	Diffusion	#2
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 11	The finite difference discretization below ensures proper conservation. $\frac{u_{i+1}^{2} - u_{i-1}^{2}}{2 \Delta x}$ The advection term of the form $u_{i} \frac{u_{i+1} - u_{i-1}}{2 \Delta x}$ 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.

August 29, 2017, 23:02	Conservative schemes	#4
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 11	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. FMDenaro likes this.

August 30, 2017, 04:59		#5
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	I agree, consider the Burgers equation in divergence and non-divergence (quasi-linear) form: du/dt + d(u^2/2)/dx=d(mudu/dx)/dx (divergence form) du/dt + udu/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. arungovindneelan and selig5576 like this.

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August 29, 2017, 22:36		#3
mseka New Member Join Date: Jul 2017 Posts: 6 Rep Power: 9	So you mean all is good with the conservative form of discretization? And no oscillations occur?