implicit-explicit methods and operating splitting

selig5576 · July 7, 2017, 12:00

According to Gilbert Strang's book, Computational Science and Engineering, he defines a scheme that implicitly treats the viscous terms and explicitly treats the advections terms, i.e.

Step 1: Advection
$\frac{u^{*} - u^{n}}{dt} = - \mathbf{u} \cdot \nabla \mathbf{u}$

Step 2: Diffusion
$\frac{u^{**} - u^{*}}{dt} = \nu \nabla^{2} \mathbf{u}$

Step 3: Solve PPE (SOR method, CG, ...)
Step 4: Corrector step
$\frac{u^{n} - u^{**}}{dt} = \frac{1}{\rho} \left(\frac{P_{i+1,j} - P_{i-1,j}}{2 dx}\right)$

In effect, its a standard Chorin-Projection method but with operator splitting for the advection and diffusion terms. My question is, are there disadvantages to performing operator splitting as described? I've seen that AB2-CN IMEX method doesnt seem to decouple advection and diffusion.

FMDenaro · July 7, 2017, 12:10

There are no practical advantages in doing a splitting between convection and diffusion in the momentum equation.
The splitting can have advantage in reaction-diffusion equations where the characteristic time of the diffusion is some order of magnitude greater than that of the reaction.
Some advantage can be possible in the splitting of the Laplace operator in three 1d problems.

selig5576 · July 7, 2017, 12:18

Thank you for the info. In an unstructured grid, would operator splitting still hold? I know that the ADI method can't be applied to unstructured grids, which makes me think that applying the TDMA for each direction would not hold? Mathematically, the operator splitting is elegant.

FMDenaro · July 7, 2017, 12:38

Quote:

Originally Posted by selig5576

Thank you for the info. In an unstructured grid, would operator splitting still hold? I know that the ADI method can't be applied to unstructured grids, which makes me think that applying the TDMA for each direction would not hold? Mathematically, the operator splitting is elegant.

From a mathematical point of view you can always write the Laplacian as a product of 1D operators along x and y (and z in 3D) but on unstructured grid you have some problem in using the differential equation instead of the integral one.

July 7, 2017, 12:00	implicit-explicit methods and operating splitting	#1
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 10	According to Gilbert Strang's book, Computational Science and Engineering, he defines a scheme that implicitly treats the viscous terms and explicitly treats the advections terms, i.e. Step 1: Advection $\frac{u^{} - u^{n}}{dt} = - \mathbf{u} \cdot \nabla \mathbf{u}$ Step 2: Diffusion $\frac{u^{} - u^{}}{dt} = \nu \nabla^{2} \mathbf{u}$ Step 3: Solve PPE (SOR method, CG, ...) Step 4: Corrector step $\frac{u^{n} - u^{**}}{dt} = \frac{1}{\rho} \left(\frac{P_{i+1,j} - P_{i-1,j}}{2 dx}\right)$ In effect, its a standard Chorin-Projection method but with operator splitting for the advection and diffusion terms. My question is, are there disadvantages to performing operator splitting as described? I've seen that AB2-CN IMEX method doesnt seem to decouple advection and diffusion.

July 7, 2017, 12:10		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71	There are no practical advantages in doing a splitting between convection and diffusion in the momentum equation. The splitting can have advantage in reaction-diffusion equations where the characteristic time of the diffusion is some order of magnitude greater than that of the reaction. Some advantage can be possible in the splitting of the Laplace operator in three 1d problems. selig5576 likes this.

July 7, 2017, 12:18	Operator splitting	#3
selig5576 Senior Member Selig Join Date: Jul 2016 Posts: 213 Rep Power: 10	Thank you for the info. In an unstructured grid, would operator splitting still hold? I know that the ADI method can't be applied to unstructured grids, which makes me think that applying the TDMA for each direction would not hold? Mathematically, the operator splitting is elegant.

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