# A Lattice Boltzmann Method and Asynchronous Model Coupling for Viscoelastic Fluids
## 3 The Coupled LBM of Viscoelastic Fluids
### The Temporal Marching of Coupled Solvers  
The computing steps of the velocity and viscoelastic stress fields are arranged as follows: (I) Equation (24) are run firstly to simulate the constitutive equations, of whose results are utilized to calculate the elastic forces; (II) The velocity fields are simulated using Equation (14), and the obtainedresults are utilized to calculate the related velocity gradient; (III) The velocity and elastic stress fields are updated iteratively at each step. In this part, we mainly discuss the methods of time marching in coupled solvers. For viscoelastic fluid flows at low Re number, the coupled systems have a very huge difference of the time scales between the constitutive modes and lattice Boltzmann evolution. So, we must carefully handle with the time scale when simulating the real geometry device. One approach to time advancement in a coupled framework is to operate the solvers at the smallest scale that required by the lattice Boltzmann equation. However, the computational cost will normally be much larger than it needed. Such a type of method is displayed in Figure 1a and we named it as the fully coupled (FC) method. To improve the efficiency of simulations, the asynchronously coupled method was applied in previous work [21]. There are two time steps being used in this coupled framework. The lattice Boltzmann equation runs by the suitable time step corresponding to its physical model, but the macro constitutive equations are solver with fewer numbers of time steps (with larger time-step sizes) than the mesoscale dynamics. The named asynchronously coupled methods mean that the data exchange mode between the lattice Boltzmann equation and macro constitutive equation at every time step. On a separate note, the numerical error that was caused by the scale-separated method may be acceptable if the time step of the macro constitutive is chosen properly. This method is shown schematically in Figure 1b and we refer to it as the asynchronously coupled (AC) method. The coupled algorithm using a homogeneous time step is given, as following Algorithm 1. ```
Require:\(f_{a}(\mathbf{x},t^{n})\), \(\boldsymbol{\tau}(\mathbf{x},\widehat{t}^{n})\), \(\delta t\), \(\Delta t\), \(T\) (is a dimension less terminal time) Do \(f_{a}(\mathbf{x},t^{n}+\delta t)\ =\ f_{a}(\mathbf{x},t^{n})-\frac{1}{\lambda}[f_{a}( \mathbf{x},t^{n})-\widehat{f}_{a}^{(eq)}(\mathbf{x},t^{n})]+\delta tF_{a}\); \(F_{a}\ =\ \psi(\boldsymbol{\tau}(\mathbf{x},\widehat{t}^{n}))\); \(f_{a}(\mathbf{x},t^{n+1})\ =\ f_{a}(\mathbf{x},t^{n}\ +\ \delta t)\); \(\boldsymbol{\tau}(\mathbf{x},\widehat{t}^{n}+\Delta t)\ =\ \boldsymbol{\tau}( \mathbf{x},\widehat{t}^{n})\ +\ \Delta tL(\boldsymbol{\tau}(\mathbf{x},\widehat{t}^{n}),f_{a}( \mathbf{x},t^{n}+\delta t))\); \(\boldsymbol{\tau}(\mathbf{x},\widehat{t}^{n+1})\ =\ \boldsymbol{\tau}( \mathbf{x},\widehat{t}^{n}\ +\ \Delta t)\); \(n\ =\ n+1\); while\(n\Delta t\ \leq\ T\)
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**Algorithm 1** The coupled algorithm using a homogeneous time step. For viscoelastic fluids at low _Wi_ number, the asynchronous time marching of the coupled solvers could get steady result in simulations if the macro time step \(\Delta t\) is chosen properly. However, as _Wi_ number increases, the asynchronously coupled method possesses fairly poor numerical stability. To widen the application of the coupled lattice Boltzmann model, we propose a generalized asynchronously coupled method for time-stably advancing of the viscoelastic lattice Boltzmann method. In this method, the asynchronously coupled way adding an iterative procedure is performed, which may allow for the lattice Boltzmann solver to run with a flexible number of steps (\(N\)) before the exchange of coupling variables. The number of added steps, \(N\), gives a further role to relax the coupling instances system. We refer to this method as the generalized asynchronously coupling (GAC) method (see Figure 1c), which can extend the applicability over a wide range of _Wi_ number.