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

Jul 16, 2026

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Active Flux is a numerical method for conservation laws that combines cell averages with point values at cell interfaces, drawing ideas from finite volumes and finite differences. The work develops a theoretical justification by reformulating Active Flux within the summation-by-parts (SBP) operator framework, enabling the design of energy-stable formulations using classical or degenerate SBP operators. The analysis is carried out for the 1D scalar linear advection equation on a uniform grid with periodic boundary conditions.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages26

Reading Time1 h 6 min

Introduction

Energy stability goal

Active Flux degrees of freedom and flux approximation

SBP framework and discrete energy analysis

arXiv :2507 . 11068v2 [math .NA] 11 Jul 2026 Stability of the Active Flux Method in the Framework of Summation-by-Parts Operators Wasilĳ Barsukow1, Christian Klingenberg2, Lisa Lechner2, Jan Nordström3,4, Sigrun Ortleb5, and Hendrik Ranocha6 1Institut de Mathématiques de Bordeaux (IMB), CNRS UMR 5251, 351 Cours de la Libération, 33405 Talence, France, wasilĳ.[barsukow@math.u-bordeaux.fr](barsukow@math.u-bordeaux.fr) 2University of Würzburg, Institute of Mathematics, Emil-Fischer-Straße 40, 97074 Würzburg, Germany 3Department of Mathematics, Linköping University, SE-581 83 Linköping, Sweden 4Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park 2006, Johannesburg, South Africa 5University of Kassel, Institute of Mathematics, Untere Königsstraße 86, 34117 Kassel, Germany 6Johannes Gutenberg University Mainz, Institute of Mathematics, Staudingerweg 9, 55128 Mainz, Germany July 14, 2026 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. Key words. Active Flux method, summation-by-parts operators, conservation laws, finite difference methods, finite volume methods AMS subject classification. 65M06, 65M20, 65M70 1. Introduction The Active Flux schemes are a class of methods introduced to solve systems of hyperbolic conservation laws (see [18–20]), an extension of Scheme V from [35] . They combine two types of degrees of freedom: cell averages and shared point values at the cell interfaces. The Active Flux method uses a globally continuous approximation, which is conceptually different to other finite volume methods. Yet, the conservative updates ofthe cell averages resemble a finite volume method and are given by the fluxes through the boundary of the cell. The Active Flux method approximates these fluxes by quadratures directly using the point values. This is in contrast to many finite volume schemes that use Riemann solvers to define the fluxes. Two ways how the point values can be updated in time have emerged. Initially (e.g., in [20, 35]), it was proposed to use a short-time (approximate) solution of the initial-value problem (IVP) of the conservation law. Such an approach is appropriate for many equations (see [7, 18] for scalar conservation laws and for hyperbolic systems of conservation laws in one spatial dimension) and the resulting method is a one-stage method. The solution of the IVP naturally includes upwinding which is helpful for stability upon explicit integration in time. Since it is quite challenging to find short-time (approximate) third-order accurate solutions for multi-dimensional systems of conservation laws, it was proposed in [5, 6] to complement a semi-discrete version of the cell average update with an ordinary differential equation (ODE) for the point value and to integrate both in time using a standard Runge-Kutta method [1–3] . The aim of this paper is to provide the first energy stability analysis of a semi-discrete Active Flux scheme using the framework of summation-by-parts (SBP) operators. The Active Flux method distinguishes itself through its low dissipation and dispersion property ([57]) . Especially in multi

FAQs

What is the Active Flux method based on, and how does it represent unknowns?

It is a conservation-law numerical method using a mix of cell averages and shared point values at cell interfaces. These point values are used to approximate fluxes via quadratures.

How does this paper connect Active Flux to summation-by-parts (SBP) operators?

It reformulates the semi-discrete Active Flux scheme within the SBP operator framework, using SBP properties to transfer continuous energy stability reasoning to the discrete level.

For what model problem and setting is the stability analysis presented?

The paper presents the analysis for the one-dimensional scalar linear advection equation on a uniform grid with periodic boundary conditions.

Jul 16, 2026

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