Localized Model-Upscaling (LMU) and Macro-guided micro (MGM) methods

Localized Model-Upscaling (LMU) method

The localized model-upscaling (LMU) method consists in coupling a perturbation model and its asymptotic limit model (when the perturbation parameter is sent to zero) through a transition zone. In the transition zone, the solution is decomposed into a ‘microscopic fraction’ (described by the perturbation problem) and a ‘macroscopic one’ (described by the limit problem). The coupling between the two components arises through source terms which can be rigorously derived from the asymptotic limit.

This strategy avoids the arbitrary definition of coupling or matching conditions, unlike standard coupling methods. The transition zone is defined by means of a cut-off function, which offers an easy adaptivity in time, and does not require to match the mesh to the geometry of the transition region. The topology of the regions where either model is used can be adaptively changed, according to appropriate 'model refinement' criteria. Therefore, the model can be locally upgraded from the macroscopic to the microscopic one, leading to a localized 'model-upscaling' methodology.

The picture below depicts the flow of a rarefied gas past a cylinder, when the model is locally upscaled from the Euler equations (far away from the cylinder) to the Boltzmann equation (in the vicinity of the cylinder). The LMU method is compared to a full resolution of the Boltzmann equation. The picture displays contour plots of the density (dots = Boltzmann eq., solid = LMU method). Both the Euler and Boltzmann equations are solved by means of deterministic finite-volume methods (where a Discrete Velocity Model is used as velocity discretization of the Boltzmann equation). In this case, the transition region is kept fixed.

The following picture displays an example where an adaptively modified transition region is used. A non-stationary one-dimensional shock is shown. On the left picture, the density as a function of position at a given time is displayed. The insert shows a blow-up of the solution in the shock region and provides the solution of the full kinetic model and that of a hydrodynamic solver for comparison. The solution provided provided by the LMU method is very close to that of the full kinetic model while that of the hydrodynamic solver is quite far off.

The right picture diplays the transition function which is used, as well as the local Knudsen number and the heat flux (Q), which are two quantities used in the definition of the transition region. According to the values of these parameters which permit to determine if the flow is of kinetic or hydrodynamic nature, a new transition region can be created. The LMU method can treat any possible kind of geometric or topological change of the kinetic and fluid zones. Again, hree, both the Euler and Boltzmann equations are solved by means of deterministic finite-volume methods.

The most widely used method to solve the Boltzmann equation is the so-called Direct Simulation Monte-Carlo (DSMC) method. In general, the coupling between the DSMC method for the Boltzmann equation with the Euler equations is considered a difficult problem, because the stochastic noise generated by the Monte-Carlo method considerably degrades the accuracy of the hydrodynamic calculation.

Still, the LMU method, used in conjunction with the Macro-Guided Micro (MGM) methodology, is able to handle this situation. The picture below shows the density for an unsteady shock test, at Knudsen =10^-2 (standard DSMC method (right) and 'LMU-DSMC' method (left)). A significant improvement in the accuracy of the method can be observed.

Macro-Guided Micro (MGM) method

This method relies on the joint solution of the microscopic and the macroscopic scales. The postulate is that it is possible to improve the accuracy on the microscopic scale if one uses corrective information collected from the macroscopic scale. Usually, the simulation of the macroscopic scale is more accurate and less expensive than that of the microscopic one and there is a net gain either in computer time or in accuracy for the microscopic scale simulation.

We exploit this idea for rarefied gas dynamics problems. We simultaneously evolve the Boltzmann equation and its moment equations up to a certain order. To evaluate the fluxes in the moment equations, we use the solution of the Boltzmann equation. After each time step, the solution of the Boltzmann equation is modified to match the solution of the moment equations. This moment matching procedures reduces the noise inherent to standard statistical methods such as the DSMC method. In addition, in the regions where the flow is at equilibrium, the LMU and MGM methods can be combined and in the 'fluid regions', only the fluid model is solved.

The picture below display DSMC computations (mass density as a function of position for an unsteady shock moving to the right) of the Boltzmann equation without (top and with (bottom) the moment-guiding procedure. As obvious from the figure, the MGM procedure significantly reduces the statistical fluctuations. The Knudsen number is 0.01.

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