Philipp Öffner

Jan Glaubitz, Henry Haase, Philipp Öffner et al.

Function-space summation-by-parts (FSBP) operators enable conservative and energy-stable numerical methods for hyperbolic conservation laws based on general, non-polynomial approximation spaces. Recent works show that using generalized Gaussian quadrature significantly reduces the number of grid points required compared to existing constructions that have mostly focused on equidistant grids. In this paper, we compare open and closed FSBP operators constructed with generalized Gaussian quadratures and apply them to numerically solve hyperbolic conservation laws. Furthermore, to support open node distributions, we extend the FSBP framework by introducing function-space exact extrapolation operators and operationalize them in numerical schemes for solving hyperbolic conservation laws. Our numerical experiments include the one-dimensional linear advection, non-viscous Burgers, and compressible Euler equations of gas dynamics. We observe that applying FSBP operators in numerical schemes can improve efficiency and accuracy. Notably, we demonstrate these advantages in more challenging time-dependent settings compared to other recent works on Gaussian FSBP operators.

Hendrik Ranocha, Jan Glaubitz, Philipp Öffner et al.

The correction procedure via reconstruction (CPR, also known as flux reconstruction) is a framework of high order semidiscretisations used for the numerical solution of hyperbolic conservation laws. Using a reformulation of these schemes relying on summation-by-parts (SBP) operators and simultaneous approximation terms (SATs), artificial dissipation / spectral viscosity operators are investigated in this first part of a series. Semidiscrete stability results for linear advection and Burgers' equation as model problems are extended to fully discrete stability by an explicit Euler method. As second part of this series, Glaubitz, Ranocha, Öffner, and Sonar (Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering, 2016) investigate connections to modal filters and their application instead of artificial dissipation.

Jan Glaubitz, Hendrik Ranocha, Philipp Öffner et al.

A recently introduced framework of semidiscretisations for hyperbolic conservation laws known as correction procedure via reconstruction (CPR, also known as flux reconstruction) is considered in the extended setting of summation-by-parts (SBP) operators using simultaneous approximation terms (SATs). This reformulation can yield stable semidiscretisations for linear advection and Burgers' equation as model problems. In order to enhance these properties, modal filters are introduced to this framework. As a second part of a series, the results of Ranocha, Glaubitz, Öffner, and Sonar ("Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation", 2016) concerning artificial dissipation / spectral viscosity are extended, yielding fully discrete stable schemes. Additionally, a new adaptive strategy to compute the filter strength is introduced and different possible applications of modal filters are compared both theoretically and numerically.

Thomas Eiter, Jan Giesselmann, Robert Lasarzik et al.

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.

Philipp Öffner, Per Pettersson, Andrew R. Winters

Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. %In particular, Bender and Öffner developed entropy-conservative discontinuous Galerkin (DG) methods to solve the stochastic shallow water equations in an stochastic Galerkin framework using Roe variable transformation, while Dai, Epshteyn and collaborators proposed second-order, energy-stable and well-balanced schemes for the same class of problems with a specific projection step used inside the Galerkin projection together with high-order quadrature rules and a time-step restriction. In this paper, we provide a comprehensive comparison of the two methodologies mentioned, focusing on their theoretical properties and practical implementation aspects. We highlight shared foundational concepts and key differences of both approaches, with a particular focus on the selection of basis functions in the stochastic domain. As a highlight, we show that under specific conditions, the two formulations align, offering a unified framework that connects these distinct approaches. From our theoretical findings, we extend the development of high-order entropy conservative DG methods for the one-dimensional stochastic Galerkin shallow equations to two space dimensions; constructing entropy conservative two-point fluxes via primitive variables instead of entropy variables and applying it in our high-order DG setting. In numerical simulations, we verify and support our theoretical findings of a well-balanced and entropy-stable DG scheme which can be used to solve geophyiscal fluid flows with uncertainty.

Jaya Agnihotri, Philipp Öffner

We present a convergence analysis of a finite volume (FV) scheme for the multicomponent compressible Euler system in the framework of dissipative weak (DW) solutions. DW solutions were introduced as a generalized solution framework in computational fluid dynamics and have recently gained considerable attention. They extend the well-known Lax Equivalence Theorem to nonlinear settings, meaning that if a numerical scheme is both consistent and stable, it will also converge. The FV scheme under consideration preserves key physical properties of the fluid mixture, in particular, positivity of partial densities, pressure, and temperature. Using uniform stability bounds and consistency estimates, we prove that the numerical solutions converge in the framework of DW solutions of the multicomponent Euler system. Applying the relative entropy method and the weak-strong uniqueness principle, we further show that the approximate solutions converge strongly to the classical solution as long as it exists. Numerical experiments confirm the theoretical results, not only for low-order FV methods but also through extended numerical investigations of a higher-order, structure-preserving discontinuous Galerkin scheme.

Jan Giesselmann, Philipp Öffner, Robert Sauerborn

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.

René Goertz, Philipp Öffner

We consider the well-known method of least squares on an equidistant grid with $N+1$ nodes on the interval $[-1,1]$. We investigate the following problem: For which ratio $N/n$ and which functions, do we have pointwise convergence of the least square operator ${LS}_n^N:\mathcal{C}\left[-1,1\right]\rightarrow\mathcal{P}_n$? To solve this problem we investigate the relation between the Jacobi polynomials $P_k^{α,β}$ and the Hahn polynomials $Q_k\left(\cdot;α,β,N\right)$. Thereby we describe the least square operator ${LS}_n^N$ by the expansion of a function by Hahn polynomials. In particular we present the following result: The series expansion $\sum_{k=0}^n{\hat{f} Q_k}$ of a function $f$ by Hahn polynomials $Q_k$ converges pointwise, if the series expansion $\sum_{k=0}^n{\hat{f} P_k}$ of the function $f$ by Jacobi polynomials $P_k$ converges pointwise and if ${n^4}/N\rightarrow 0$ for $n,N\rightarrow\infty$. Furthermore we obtain the following result: Let $f\in\left\{g\in\mathcal{C}^1\left[-1,1\right]:g^\prime\in\mathcal{BV}\left[-1,1\right]\right\}$ and let $(N_n)_{n}$ be a sequence of natural numbers with ${n^4}/{N_n}\rightarrow 0$. Then the least square method ${LS}_n^{N_n}[f]$ converges for each $x\in[-1,1]$.

René Goertz, Philipp Öffner

We consider in this paper the Hahn polynomials and their application in numerical methods. The Hahn polynomials are classical discrete orthogonal polynomials. We analyse the behaviour of these polynomials in the context of spectral approximation of partial differential equations. We study series expansions $u=\sum_{n=0}^\infty \hat{u}_n ϕ_n$, where the $ϕ_n$ are the Hahn polynomials. We examine the Hahn coefficients and proof spectral accuracy in some sense. We substantiate our results by numericals tests. Furthermore we discuss a problem which arise by using the Hahn polynomials in the approximation of a function $u$, which is linked to the Runge phenomenon. We suggest two approaches to avoid this problem. These will also be the motivation and the outlook of further research in the application of discrete orthogonal polynomials in a spectral method for the numerical solution of hyperbolic conservation laws.