arXiv subjects

Robert Sauerborn

Publications and source records attributed to Robert Sauerborn.

3 recordsLinked to original sources

arXiv2026

Convergence of a Finite Volume Scheme for the Navier-Stokes-Korteweg Model via Dissipative Solutions

We propose a concept of dissipative weak (DW) solutions for the Navier-Stokes-Korteweg (NSK) system and prove conditional convergence of a structure-preserving finite volume scheme towards such a solution. DW solutions provide a generalized solution concept in computational fluid dynamics and have recently attracted significant attention. They provide an extension of the famous Lax Equivalence Theorem to nonlinear problems, i.e. consistency and stability of a numerical scheme imply convergence. Our work builds on recent advances where convergence towards DW solutions of structure-preserving schemes has been established for the Euler and Navier-Stokes equations. Indeed, we prove convergence of a recently proposed FV scheme by leveraging its conservation and dissipation properties as well as its consistency.

math.NA↗

arXiv2026

Convergence analysis for a finite-volume scheme for the Euler- and Navier-Stokes-Korteweg system via energy-variational solutions

We consider a structure-preserving finite-volume scheme for the Euler-Korteweg (EK) and Navier-Stokes-Korteweg (NSK) equations. We prove that its numerical solutions converge to energy-variational solutions of EK or NSK under mesh refinement. Energy-variational solutions constitute a novel solution concept that has recently been introduced for hyperbolic conservation laws, including the EK system, and which we extend to the NSK model. Our proof is based on establishing uniform estimates following from the properties of the structure-preserving scheme, and using the stability of the energy-variational formulation under weak convergence in the natural energy spaces.

math.NA↗

arXiv2026

A Structure Preserving Finite Volume Scheme for the Navier-Stokes-Korteweg Equations

We present a semi-discrete finite volume scheme for the local NavierStokes-Korteweg and Euler-Korteweg systems. Our scheme is applicable for equidistant Cartesian meshes in one and two space dimensions. In contrast to other works, which employ, for example, hyperbolic approximations of the equations or auxiliary-variable approaches leading to extended systems, our scheme operates directly on the original system. We prove that it conserves mass and momentum and is energy stable. Numerical experiments complement our theoretical findings, showing that the scheme is convergent of order one if employed with explicit or implicit time discretisation.

math.NA↗