Implicit Time Integration for Euler Equations

maHein · April 20, 2015, 07:50

Hello everybody,

I would like to implement a implicit time integration for Euler equations. For the explicit case, this is pretty straight forward:

$\frac{V_i}{\Delta t_i} ( \vec{W}_i^{n+1} - \vec{W}_i^{n}) + \sum_{j \in N(i)} \vec{F}_{ij}^n S_{ij} = 0$

which results in the following OpenFOAM code for the conservation of mass:

Code:

solve
(
    fvm::ddt(rho)
  + fvc::div(rhoFlux)
);

For the implicit case

$\frac{V_i}{\Delta t_i} ( \vec{W}_i^{n+1} - \vec{W}_i^{n}) + \sum_{j \in N(i)} \vec{F}_{ij}^{n+1} S_{ij} = 0$

one can linearize the flux at time n+1 using Taylor's series expansion

$\vec{F}_{ji}^{n+1} \approx \vec{F}^n_{ij} + \left( \frac{\partial \vec{F}}{\partial \vec{W}} \right)_{ij} \Delta \vec{W}^n_{ij}$

with the flux Jacobian $\frac{\partial \vec{F}}{\partial \vec{W}}$ calculated using finite difference differentiation. This results in

$\frac{V_i}{\Delta t_i} ( \vec{W}_i^{n+1} - \vec{W}_i^{n}) + \sum_{j \in N(i)} \frac{\partial \vec{F}}{\partial \vec{W}} (\vec{W}_{ij}^{n+1} - \vec{W}_{ij}^{n}) S_{ij} = -Res_i^n$

with the term of the right hand side being the fluxes.

My OpenFOAM implementation of this likes the following for the conservation of mass

Code:

solve
(
    fvm::ddt(rho)
  + fvm::div(jacobianRho, rho)
  - fvc::div(jacobianRho, rho)
 ==
  - fvc::div(rhoFlux)
);

Unfortunatelly, this implementation is not stable and diverges very quickly after a few iterations.

Is my implementation incorrect or where is my mistake in this? Is there maybe a better way to realize this method in OpenFOAM?

Thanks for your help!

Martin

ssss · April 20, 2015, 13:40

So I think I need more information to know what you are trying to do, are you only solving the density equation? Are you solving the Navier-Stokes?

Bare in mind that OpenFOAM uses the Rhie-Chow interpolation method and some assumptions about which terms are taken as explicit and which terms are taken as implicit.

Maybe you can obtain the results you are looking using the correct schemes

April 20, 2015, 07:50	Implicit Time Integration for Euler Equations	#1
maHein Member Join Date: Jun 2012 Posts: 76 Rep Power: 13	Hello everybody, I would like to implement a implicit time integration for Euler equations. For the explicit case, this is pretty straight forward: $\frac{V_i}{\Delta t_i} ( \vec{W}_i^{n+1} - \vec{W}_i^{n}) + \sum_{j \in N(i)} \vec{F}_{ij}^n S_{ij} = 0$ which results in the following OpenFOAM code for the conservation of mass: Code: solve ( fvm::ddt(rho) + fvc::div(rhoFlux) ); For the implicit case $\frac{V_i}{\Delta t_i} ( \vec{W}_i^{n+1} - \vec{W}_i^{n}) + \sum_{j \in N(i)} \vec{F}_{ij}^{n+1} S_{ij} = 0$ one can linearize the flux at time n+1 using Taylor's series expansion $\vec{F}_{ji}^{n+1} \approx \vec{F}^n_{ij} + \left( \frac{\partial \vec{F}}{\partial \vec{W}} \right)_{ij} \Delta \vec{W}^n_{ij}$ with the flux Jacobian $\frac{\partial \vec{F}}{\partial \vec{W}}$ calculated using finite difference differentiation. This results in $\frac{V_i}{\Delta t_i} ( \vec{W}_i^{n+1} - \vec{W}_i^{n}) + \sum_{j \in N(i)} \frac{\partial \vec{F}}{\partial \vec{W}} (\vec{W}_{ij}^{n+1} - \vec{W}_{ij}^{n}) S_{ij} = -Res_i^n$ with the term of the right hand side being the fluxes. My OpenFOAM implementation of this likes the following for the conservation of mass Code: solve ( fvm::ddt(rho) + fvm::div(jacobianRho, rho) - fvc::div(jacobianRho, rho) == - fvc::div(rhoFlux) ); Unfortunatelly, this implementation is not stable and diverges very quickly after a few iterations. Is my implementation incorrect or where is my mistake in this? Is there maybe a better way to realize this method in OpenFOAM? Thanks for your help! Martin

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April 20, 2015, 13:40		#2
ssss Senior Member anonymous Join Date: Aug 2014 Posts: 205 Rep Power: 12	So I think I need more information to know what you are trying to do, are you only solving the density equation? Are you solving the Navier-Stokes? Bare in mind that OpenFOAM uses the Rhie-Chow interpolation method and some assumptions about which terms are taken as explicit and which terms are taken as implicit. Maybe you can obtain the results you are looking using the correct schemes