arXiv subjects

Christian Klingenberg

Publications and source records attributed to Christian Klingenberg.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic preserving IMEX Active Flux schemes for a hyperbolic approximation of the Korteweg-de Vries equation

We present a theoretical and numerical analysis of novel IMEX Active Flux schemes for a hyperbolic approximation of the Korteweg-de Vries (KdV) equation. For this approximation, it is important that the numerical schemes recover the limit equations (asymptotic preserving) and retain their order of accuracy in the limit (asymptotically accurate). Using formal asymptotic analysis and numerical experiments, we show that an Active Flux spatial discretisation (closely related to the standard semi-discrete Active Flux scheme), combined with an implicit-explicit Runge-Kutta scheme, is asymptotic preserving but not asymptotically accurate. We therefore propose alternative spatial discretisations that differ in the update equations of the point values. First, we derive a scheme by rescaling and eliminating terms in the upwind matrices. Second, we develop a second-order scheme by using central discretisation for the terms treated implicitly in time and present two modifications of it to third-order. Finally, we formulate a scheme with predefined upwind directions that are not chosen based on characteristics. We show that the alternative schemes are not only asymptotic preserving but also asymptotically accurate, and compare their accuracy, stability, and energy conservation numerically.

math.NA↗

arXiv2026

On Enhancing the Dissipative Behavior of Active Flux Advection Schemes

In this work, the traditional third-order Active Flux advection scheme is modified by reformulating the method and introducing four additional parameters R, S, T, and U that allow the numerical dissipation and dispersion of the scheme to be adjusted. The influence of these parameters is studied using Fourier analysis and parameter choices that lead to schemes with improved dissipative properties are identified. These improvements are validated by numerical experiments.

math.NA↗

arXiv2026

An adaptive dynamical low-rank optimizer for solving kinetic parameter identification problems

The numerical solution of parameter identification problems for kinetic equations can exhibit high computational and memory costs. In this paper, we propose a dynamical low-rank scheme for the reconstruction of the scattering parameter in the radiative transfer equation from a number of macroscopic time-independent measurements. We first work through the PDE constrained optimization procedure in a continuous setting and derive the adjoint equations using a Lagrangian reformulation. For the scattering coefficient, a periodic B-spline approximation is introduced and a gradient descent step for updating its coefficients is formulated. After the discretization, a dynamical low-rank approximation (DLRA) is applied. We make use of the rank-adaptive basis update & Galerkin integrator and a line search approach for the adaptive refinement of the gradient descent step size and the DLRA tolerance. We show that the proposed scheme significantly reduces both memory and computational cost. Numerical results computed with different initial conditions validate the accuracy and efficiency of the proposed DLRA scheme compared to solutions computed with a full solver.

math.NA↗

arXiv2026

A compact high-order and positivity-preserving active flux method for compressible Navier--Stokes equations on Cartesian meshes

This paper develops a compact fourth-order positivity-preserving active flux (AF) method for the one- and two-dimensional compressible Navier--Stokes equations on Cartesian meshes. The method retains the cell averages and shared point values of the standard third-order AF method as its degrees of freedom. To avoid the order reduction that can arise when diffusion is discretized using operators from the standard third-order AF method, while maintaining compactness, the divergence of the viscous flux is discretized directly using compact fourth-order operators. For the inviscid part, incorporating a downwind point value into the biased stencil yields fourth-order accuracy. A monolithic flux limiting blends high-order total numerical fluxes with low-order positivity-preserving counterparts, treating the inviscid and viscous fluxes jointly while maintaining local conservation. Together with a scaling limiter for point values, this procedure preserves density and pressure positivity for both cell averages and point values. Numerical experiments demonstrate fourth-order convergence, positivity preservation, and accurate resolution of shocks and viscous flow structures. For the two-dimensional viscous shock tube, a comparison with a discontinuous Galerkin method shows improved computational efficiency in resolving complex interaction of shock waves and boundary layers.

math.NA↗

arXiv2026

Semi-discrete Active Flux method for two-dimensional hyperbolic systems with Coriolis source terms

We investigate the semi-discrete Active Flux numerical method on Cartesian grids applied to two-dimensional linear and nonlinear shallow water equations with Coriolis source terms. These hyperbolic systems admit non-trivial stationary solutions governed by a geostrophic equilibrium. In this setting, we analyze the stationarity-preserving properties of our semi-discrete Active Flux formulation. To do so for the linear system, we apply a discrete spatial Fourier transform to the semi-discrete method and analyze the resulting evolution matrix. We show that its discrete kernel is a non-trivial discretization of the geostrophic equilibrium, which implies that the method preserves a discrete geostrophic equilibrium state. This is an inherent feature of Active Flux and is achieved without any additional modifications. Numerical experiments confirm the method's ability to maintain geostrophic stationary states for the linear system. Additionally, we numerically investigate the performance of the Active Flux method for the nonlinear shallow water system.

math.NA↗

arXiv2026

Local sensitivity-preserving random data down-sampling for experimental design

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 preserve 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.

math.NA↗

arXiv2026

Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators

The Active Flux method is a numerical method for conservation laws using a combination of cell averages and point values as independent degrees of freedom, based on ideas from finite volumes and finite differences. This unusual mix has been shown to work well in many situations. We expand the theoretical justifications of the Active Flux method by analyzing it from the point of view of summation-by-parts (SBP) operators, which are routinely used to analyze finite difference, finite volume, and finite element schemes. We investigate in what type of setting the Active Flux method can be formulated using classical or degenerate SBP operators, yielding a first and novel approach for showing the energy stability of the Active Flux method. We present the analysis for the one-dimensional scalar linear advection equation with periodic boundary conditions on a uniform grid.

math.NA↗

arXiv2026

A stable multiplicative dynamical low-rank discretization for the linear Boltzmann-BGK equation

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.

math.NA↗

arXiv2026

An energy stable and conservative multiplicative dynamical low-rank discretization for the Su-Olson problem

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.

math.NA↗

arXiv2026

On the equivalence of semi-discrete Active Flux and Discontinuous Galerkin methods and a comparison of their performance

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.

math.NA↗

arXiv2026

A low-dissipation central scheme for ideal MHD

Central schemes for conservation laws are Riemann solver free methods which are simple and easy to implement. In recent work for Euler equations [Kurganov & Xin, J. Sci. Comput., 96:56, 2023] their accuracy has been enhanced in terms of better resolution of contact waves. In this paper, we extend this low dissipation central upwind method to the ideal MHD system in one- and two-dimensions. In the two-dimensional case, we separate the variables into two groups: hydrodynamic and magnetic, which are stored at cell centers and faces, respectively. For the the hydrodynamic variables, we apply the low dissipation central upwind scheme while for the magnetic variables, a constrained transport method is used which maintains the divergence-free property of the magnetic field. The time integration is performed with third order strong stability preserving Runge-Kutta scheme. To validate the proposed scheme, we apply this method to several challenging test cases. The results show that the LDCU correction term plays a useful role at the contact discontinuity and enhances the resolution of waves. We also observe experimental second-order accuracy for smooth solutions and the divergence-free condition is maintained to machine precision.

math.NA↗

arXiv2026

Maximal turbulence as a selection criterion for measure-valued solutions

The quest for a good solution concept for the partial differential equations (PDEs) arising in mathematical fluid dynamics is an outstanding open problem. An important notion of solutions are the measure-valued solutions. It is well known that for many PDEs there exists a multitude of measure-valued solutions even if admissibility criteria like an energy inequality are imposed. Hence in recent years, people have tried to select the relevant solutions among all admissible measure-valued solutions or at least to rule out some solutions which are not relevant. In this paper another such criterion is studied. In particular, we aim to select generalized Young measures which are ``maximally turbulent''. To this end, we look for maximizers of a certain functional, namely the variance, or more precisely, the Jensen defect of the energy. We prove existence of such a maximizer and we show that its mean value and total energy is uniquely determined. Our theory is carried out in a very general setting which may be applied in many situations where maximally turbulent measures shall be selected among a set of generalized Young measures. Finally, we apply this general framework to the incompressible and the isentropic compressible Euler equation. Our criterion of maximal turbulence is plausible and leads to existence and uniqueness in a certain sense (in particular, the mean value and the total energy of different maximally turbulent solutions coincide).

math.AP↗

arXiv2026

BGK model for rarefied gas in a bounded domain

We study the Bathnagar-Gross-Krook (BGK) equation in a smooth bounded domain featuring a diffusive reflection boundary condition with general collision frequency. We prove that the BGK equation admits a unique global solution with an exponential convergence rate if the initial condition is a small perturbation around the global Maxwellian in the $L^\infty$ space. For the proof, we utilize the dissipative nature from the linearized BGK operator and establish an $L^2$ coercive estimate. Next, we derive the a priori estimate by obtaining an $L^\infty$ bound on the nonlinear operator; this requires a delicate analysis to manage its intrinsic nonlinear structure. Finally, we establish the $L^\infty$ stability estimate and introduce sequential arguments for the nonlinear BGK operator, thereby concluding both well-posedness and positivity.

math.AP↗

arXiv2025

A generalized Active Flux method of arbitrarily high order in two dimensions

The Active Flux method can be seen as an extended finite volume method. The degrees of freedom of this method are cell averages, as in finite volume methods, and in addition shared point values at the cell interfaces, giving rise to a globally continuous reconstruction. Its classical version was introduced as a one-stage fully discrete, third-order method. Recently, a semi-discrete version of the Active Flux method was presented with various extensions to arbitrarily high order in one space dimension. In this paper we extend the semi-discrete Active Flux method on two-dimensional Cartesian grids to arbitrarily high order, by including moments as additional degrees of freedom (hybrid finite element--finite volume method). The stability of this method is studied for linear advection. For a fully discrete version, using an explicit Runge-Kutta method, a CFL restriction is derived. We end by presenting numerical examples for hyperbolic conservation laws.

math.NA↗

arXiv2025

Mitigating numerical dissipation in simulations of subsonic turbulent flows

Magnetohydrodynamic (MHD) simulations of subsonic (Mach number~$<1$) turbulence are crucial to our understanding of several processes including oceanic and atmospheric flows, the amplification of magnetic fields in the early universe, accretion discs, and stratified flows in stars. In this work, we demonstrate that conventional numerical schemes are excessively dissipative in this low-Mach regime. We demonstrate that a new numerical scheme (termed `USM-BK' and implemented in the FLASH MHD code) reduces the dissipation of kinetic and magnetic energy, constrains the divergence of magnetic field to zero close to machine precision, and resolves smaller-scale structure than other, more conventional schemes, and hence, is the most accurate for simulations of low-Mach turbulent flows among the schemes compared in this work. We first compare several numerical schemes/solvers, including Split-Roe, Split-Bouchut, USM-Roe, USM-HLLC, USM-HLLD, and the new USM-BK, on a simple vortex problem. We then compare the schemes/solvers in simulations of the turbulent dynamo and show that the choice of scheme affects the growth rate, saturation level, and viscous and resistive dissipation scale of the dynamo. We also measure the numerical kinematic Reynolds number (Re) and magnetic Reynolds number (Rm) of our otherwise ideal MHD flows, and show that the new USM-BK scheme provides the highest Re and comparable Rm amongst all the schemes compared.

physics.flu-dyn↗

arXiv2025

An asymptotic-preserving active flux scheme for the hyperbolic heat equation in the diffusive scaling

The Active Flux (AF) method is a compact, high-order finite volume scheme that enhances flexibility by introducing point values at cell interfaces as additional degrees of freedom alongside cell averages. The method of lines is employed here for temporal discretization. A common approach for updating point values relies on the Jacobian Splitting (JS) method, which incorporates upwinding. A key advantage of the AF method over standard finite volume schemes is its structure-preserving property, motivating the investigation of its asymptotic-preserving (AP) behavior in the diffusive scaling. We show that the JS-based AF method without any modification is AP for solving the hyperbolic heat equation, in the sense that the limit scheme is a discretization of the limit heat equation. We use formal asymptotic analysis, discrete Fourier analysis, and numerical experiments to illustrate our findings.

math.NA↗

arXiv2025

Active flux for ideal magnetohydrodynamics: A positivity-preserving scheme with the Godunov-Powell source term

The Active Flux (AF) is a compact, high-order finite volume scheme that allows more flexibility by introducing additional point value degrees of freedom at cell interfaces. This paper proposes a positivity-preserving (PP) AF scheme for solving the ideal magnetohydrodynamics, where the Godunov-Powell source term is employed to deal with the divergence-free constraint. For the evolution of the cell average, apart from the standard conservative finite volume method for the flux derivative, the nonconservative source term is built on the quadratic reconstruction in each cell, which maintains the compact stencil in the AF scheme. For the point value update, the local Lax-Friedrichs (LLF) flux vector splitting is adopted for the flux derivative, originally proposed in [Duan, Barsukow, and Klingenberg, SIAM Journal on Scientific Computing, 47(2), A811--A837, 2025], and a central difference is used to discretize the divergence in the source term. A parametrized flux limiter and a scaling limiter are presented to preserve the density and pressure positivity by blending the AF scheme with the first-order PP LLF scheme with the source term. To suppress oscillations, a new shock sensor considering the divergence error is proposed, which is used to compute the blending coefficients for the cell average. Several numerical tests are conducted to verify the third-order accuracy, PP property, and shock-capturing ability of the scheme. The key role of the Godunov-Powell source term and its suitable discretization in controlling divergence error is also validated.

math.NA↗

arXiv2025

Analysis of the multi-dimensional semi-discrete Active Flux method using the Fourier transform

The degrees of freedom of Active Flux are cell averages and point values along the cell boundaries. These latter are shared between neighbouring cells, which gives rise to a globally continuous reconstruction. The semi-discrete Active Flux method uses its degrees of freedom to obtain Finite Difference approxi\-mations to the spatial derivatives which are used in the point value update. The averages are updated using a quadrature of the flux and making use of the point values as quadrature points. The integration in time employs standard Runge-Kutta methods. We show that this generalization of the Active Flux method in two and three spatial dimensions is stationarity preserving for linear acoustics on Cartesian grids, and present an analysis of numerical diffusion and stability.

math.NA↗