Kai Schneider

Romain Nguyen van yen, Dmitry Kolomenskiy, Kai Schneider

We report the results of a detailed study of the spectral properties of Laplace and Stokes operators, modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, $η$, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of $η$, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed $η$, we find that only the part of the spectrum corresponding to eigenvalues $λ\lesssim η^{-1}$ approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of $η$ and $λ$. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision $O(η)$, Navier slip boundary conditions with slip length equal to $\sqrtη$. Moreover, for a given discretization, we show that there exists a value of $η$, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier-Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

Ralf Deiterding, Margarete O. Domingues, Sonia M. Gomes et al.

We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in space with second order shock-capturing, and explicit time integration either with or without local time-stepping. The two methods are benchmarked for the compressible Euler equations in Cartesian geometry. As test cases a 2D Riemann problem, Lax-Liu 6, and a 3D ellipsoidally expanding shock wave have been chosen. We compare and assess their computational efficiency in terms of CPU time and memory requirements. We evaluate the accuracy by comparing the results of the adaptive computations with those obtained with the corresponding FV scheme using a regular fine mesh. We find that both approaches yield similar trends for CPU time compression for increasing number of refinement levels. MR exhibits more efficient memory compression than AMR and shows slightly enhanced convergence; however, a larger absolute overhead is measured for the tested codes.

Mostafa Bendahmane, Raimund Bürger, Ricardo Ruiz Baier et al.

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.

Beata Zorawski, Shubhaditya Burela, Philipp Krah et al.

This paper presents a neural network-based methodology for the decomposition of transport-dominated fields using the shifted proper orthogonal decomposition (sPOD). Classical sPOD methods typically require an a priori knowledge of the transport operators to determine the co-moving fields. However, in many real-life problems, such knowledge is difficult or even impossible to obtain, limiting the applicability and benefits of the sPOD. To address this issue, our approach estimates both the transport and co-moving fields simultaneously using neural networks. This is achieved by training two sub-networks dedicated to learning the transports and the co-moving fields, respectively. Applications to synthetic data and a wildland fire model illustrate the capabilities and efficiency of this neural sPOD approach, demonstrating its ability to separate the different fields effectively.