Research Highlight 05.26
04.05.2026Acom
The accurate simulation of microchannel gas flows aids in the design and development of microscopic devices. One crucial aspect of theses flows is, that compared to macroscopic channel flows, gas particles travel significant distances, with respect to the width of the channel, between collisions. Hence, the relative lack of collisions is insufficient to maintain a thermal equilibrium, leading to pronounced thermal nonequilibrium effects, which the the Navier-Stokes-Fourier equations fail to reproduce. This shortcoming of classical models necessitates extended models, such as moment equations, providing higher model-accuracy at the cost of higher computational effort. Employing these models is particularly important near curved boundaries, where large local curvature of the boundary geometry induces an elevated thermal nonequilibrium as opposed to the rest of the flow.
In our recent research, we explore a model-adaptive DG scheme for a nested hierarchy of linearized moment equations. To goal is to increase computational efficiency by keeping the model as coarse as possible while increasing its complexity only in local regions of the computational domain, where the increased accuracy is required to maintain an overall target accuracy. To accomplish this, each individual element of the discretization keeps track of its model level from the model hierarchy and neighboring elements with mismatching levels are coupled by means of two half Riemann problems. The scheme is then embedded into a standard Solve–Estimate–Mark–Refine (SEMR) loop, sketched in the Figure below. Starting from an initial coarse solution, an error estimator, such as a refined solution, is used to mark elements exhibiting high error for refinement. Thereafter, the discretization is adapted and an updated solution is computed. Once the global error derived from the estimator falls below a specified tolerance, the procedure returns the final solution. By adapting only where needed, this approach significantly reduces the amount of degrees of freedom of the discretization and consequently also the time to solution. Initial results show promise and highlight future directions of improvement, such as efficient estimators and structure-preserving model-coupling. 
Contact: Matthias Geratz
This image shows the distribution of points in the unit square. Pseudo-random numbers are scattered irregularly, while quasi-random numbers cover the space more evenly.
This image shows the error for different moments of the distribution function (energy and stress) during relaxation from a non-equilibrium state to equilibrium. The results demonstrate that quasi-random numbers achieve faster convergence and thus require fewer particles for the same accuracy.
The figure shows relative error of the next higher moment for a range velocity shifts in the bimodal test case. The left part of the figure shows the relative error for even number of moments while the right one shows odd cases. Different plot markers with colors represent the closure. Note, that the Gramian and extended Gramian closure are defined in the even and odd case differently.
This figure presents the proposed WB method (top row) against the non-well-balanced (NWB) variation (bottom row) and their abilities (or lack thereof) to capture a small perturbation of a steady-state. It is clear that the WB method captures the proper structure of the solution even on a coarse mesh, while the NWB method has smearing of the solution due to numerical error -- even on a refined mesh.
