Christian Klingenberg

Lena Baumann, Lukas Einkemmer, Christian Klingenberg et al.

The numerical method of dynamical low-rank approximation (DLRA) has recently been applied to various kinetic equations showing a significant reduction of the computational effort. In this paper, we apply this concept to the linear Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) equation which due its high dimensionality is challenging to solve. Inspired by the special structure of the non-linear Boltzmann-BGK problem, we consider a multiplicative splitting of the distribution function. We propose a rank-adaptive DLRA scheme making use of the basis update & Galerkin integrator and combine it with an additional basis augmentation to ensure numerical stability, for which an analytical proof is given and a classical hyperbolic Courant-Friedrichs-Lewy (CFL) condition is derived. This allows for a further acceleration of computational times and a better understanding of the underlying problem in finding a suitable discretization of the system. Numerical results of a series of different test examples confirm the accuracy and efficiency of the proposed method compared to the numerical solution of the full system.

Lena Baumann, Lukas Einkemmer, Christian Klingenberg et al.

Computing numerical solutions of the thermal radiative transfer equations on a finely resolved grid can be costly due to high computational and memory requirements. A numerical reduced order method that has recently been applied to a wide variety of kinetic partial differential equations is the concept of dynamical low-rank approximation (DLRA). In this paper, we consider the thermal radiative transfer equations with Su-Olson closure, leading to a linearized kinetic model. For the conducted theoretical and practical considerations we use a multiplicative splitting of the distribution function that poses additional challenges in finding an energy stable discretization and deriving a hyperbolic Courant-Friedrichs-Lewy (CFL) condition. We propose such an energy stable DLRA scheme that makes use of the augmented basis update & Galerkin integrator. This integrator allows for additional basis augmentations, enabling us to give a mathematically rigorous proof of energy stability and local mass conservation. Numerical examples confirm the derived properties and show the computational advantages of the DLRA scheme compared to a numerical solution of the full system of equations.

Wasilij Barsukow, Philipp V. F. Edelmann, Christian Klingenberg et al.

Based on the Roe solver a new technique that allows to correctly represent low Mach number flows with a discretization of the compressible Euler equations was proposed in Miczek et al.: New numerical solver for flows at various mach numbers. A&A 576, A50 (2015). We analyze properties of this scheme and demonstrate that its limit yields a discretization of the continuous limit system. Furthermore we perform a linear stability analysis for the case of explicit time integration and study the performance of the scheme under implicit time integration via the evolution of its condition number. A numerical implementation demonstrates the capabilities of the scheme on the example of the Gresho vortex which can be accurately followed down to Mach numbers of ~1e-10 .

Andrea Thomann, Markus Zenk, Christian Klingenberg

We present a well-balanced finite volume solver for the compressible Euler equations with gravity where the approximate Riemann solver is derived using a relaxation approach. Besides the well-balanced property, the scheme is robust with respect to the physical admissible states. Another feature of the method is that it can maintain general stationary solutions of the hydrostatic equilibrium up to machine precision. For the first order scheme we present a well-balanced and positivity preserving second order extension using a modified minmod slope limiter. To maintain the well-balanced property, we reconstruct in equilibrium variables. Numerical examples are performed to demonstrate the accuracy, well-balanced and positivity preserving property of the presented scheme for up to 3 space dimensions.

Jonas P. Berberich, Praveen Chandrashekar, Christian Klingenberg et al.

We develop a second order well-balanced finite volume scheme for compressible Euler equations with a gravitational source term. The well-balanced property holds for arbitrary hydrostatic solutions of the corresponding Euler equations without any restriction on the equation of state. The hydrostatic solution must be known a priori either as an analytical formula or as a discrete solution at the grid points. The scheme can be applied on curvilinear meshes and in combination with any consistent numerical flux function and time stepping routines. These properties are demonstrated on a range of numerical tests.

Nabil Bedjaoui, Christian Klingenberg, Philippe G. LeFloch

We justify a Chapman-Enskog expansion for discontinuous solutions of hyperbolic conservation laws containing shock waves with small strength. Precisely, we establish pointwise uniform estimates for the difference between the traveling waves of a relaxation model and the traveling waves of the corresponding diffusive equations determined by a Chapman-Enskog expansion procedure to first- or second-order.

Wasilij Barsukow, Christian Klingenberg, Simon Krotsch

The Active Flux (AF) method employs a globally continuous approximation, like continuous Finite Element methods. This is achieved through the placement of point values at cell interfaces which are shared between adjacent cells. With, on average, K+1 degrees of freedom per cell, Active Flux achieves a polynomial approximation of degree K+1, while the Discontinuous Galerkin (DG) method uses only polynomials of degree K, i.e. one degree less with the same number of degrees of freedom. Despite all the differences, in this paper we show, however, that for linear problems in one and several dimensions as well as -- in some sense -- for nonlinear ones, semi-discrete AF and DG are the same method. We identify a mapping between their respective degrees of freedom, upon which the updates of these degrees of freedom turn out to agree. On the one hand, AF therefore seems more economical then DG for a given value of the error, and we confirm this in numerical experiments. On the other hand, this is a way to understand superconvergence of DG in a natural way, and we show how Radau polynomials and their zeros appear in the mapping between DG and AF: In the Radau points, AF "shines through" as the background high-order scheme behind DG.

Kathrin Hellmuth, Christian Klingenberg, Qin Li

The quality of numerical reconstructions for unknown parameters in inverse problems depends fundamentally on the selection of experimental data. To ensure a robust reconstruction, it is crucial to select data that are sensitive to the parameters, a property typically characterized by the conditioning of the Fisher Information Matrix (FIM). In this work, we propose a general framework for an efficient down-sampling strategy that selects experimental setups that preserves the information content of the full-data FIM. Our approach leverages matrix sketching techniques from randomized numerical linear algebra to achieve a sensitivity-preserving approximation. The method involves drawing samples from a sensitivity-informed distribution, which we execute using gradient-free ensemble sampling methods to handle potentially non-smooth or discrete design spaces. Numerical experiments demonstrate the effectiveness of this framework in selecting optimal sensor locations for a Schroedinger potential reconstruction problem.

Christian Klingenberg, Simon Krotsch, Philip Roe

In this work, the traditional third-order Active Flux advection scheme is modified by reformulating the method and introducing additional parameters. The effect of these parameters is studied, leading to schemes with improved dissipative properties. These improvements are validated by numerical experiments.

Luis Kaiser, Richard Tsai, Christian Klingenberg

In a variety of scientific and engineering domains, the need for high-fidelity and efficient solutions for high-frequency wave propagation holds great significance. Recent advances in wave modeling use sufficiently accurate fine solver outputs to train a neural network that enhances the accuracy of a fast but inaccurate coarse solver. In this paper we build upon the work of Nguyen and Tsai (2023) and present a novel unified system that integrates a numerical solver with a deep learning component into an end-to-end framework. In the proposed setting, we investigate refinements to the network architecture and data generation algorithm. A stable and fast solver further allows the use of Parareal, a parallel-in-time algorithm to correct high-frequency wave components. Our results show that the cohesive structure improves performance without sacrificing speed, and demonstrate the importance of temporal dynamics, as well as Parareal, for accurate wave propagation.