An Active Flux method for the Euler equations based on the exact acoustic evolution operator

Jul 16, 2026

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A new Active Flux method for multi-dimensional Euler equations uses additive operator splitting into acoustics and advection. The acoustic part is advanced in a locally linearized setting via the exact evolution operator, while nonlinear advection is computed with third-order accuracy using a new approximate evolution operator. Primitive variables are used for point values and reconstruction to simplify the splitting. For discontinuities, a blended bound-preserving limiter combines a priori and a posteriori ideas, yielding capability for multidimensional Riemann problems and low-Mach flow with a wide stability domain.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages42

Reading Time1 h 46 min

Introduction

Active Flux and upwinding in one dimension

Third-order accuracy beyond local linearization

Approximate evolution operators and ADER-inspired ideas

Multidimensional extensions via acoustic evolution

Challenges for nonlinear multidimensional systems

Semi-discrete Active Flux and time discretization

arXiv :2506 .03291v2 [math .NA] 11 Jul 2026 An Active Flux method for the Euler equations based on the exact acoustic evolution operator Wasilij Barsukow 1 Abstract A new Active Flux method for the multi-dimensional Euler equations is based on an additive operator splitting into acoustics and advection. The acoustic operator is solved in a locally linearized manner by using the exact evolution operator. The nonlinear advection operator is solved at third order accuracy using a new approximate evolution operator. To simplify the splitting, the new method uses primitive variables for the point values and for the reconstruction. In order to handle discontinuous solutions, a blended bound preserving limiting is used, that combines a priori and a posteriori approaches. The resulting method is able to resolve multidimensional Riemann problems as well as low Mach number flow, and has a large domain of stability. Keywords: Active Flux, Euler equations Mathematics Subject Classification (2010): 65M08, 65M70, 76M12, 35L45 1 Introduction The Active Flux method, originally introduced for one-dimensional (1-d) linear advection in [vL77], has received much attention since the pioneering works [ER11, ER13] . As degrees of freedom, it employs averages and point values at cell interfaces. These are independent, i.e. the Active Flux method consists of update equations for both the averages and the point values. The former is easy to obtain for conservation laws: since the point values are located at cell interfaces, the flux can be directly evaluated; no Riemann solvers are needed and one recognizes the continuous nature of the spatial approximation. While the update of the averages is “exact”, [i.e. as](i.e. as) accurate as the point values, it cannot incorporate upwinding, since the latter always implies some kind of additional, artificial diffusion. It is the point values which need to include the upwinding necessary for stability. The initial method from [vL77] uses characteristic tracing and a reconstruction, whose value at the foot of the characteristic is taken as the point value at the next time step. This procedure naturally includes upwinding, of course, and yields a one-stage method stable up to CFL = 1 . Generalizations of this update procedure appeared under the name of (approximate) evolution operators in e.g. [ER13, Fan17, BHKR19, Bar21a, CHLM24] . It was shown in [Bar21a] that one needs to go one order of accuracy beyond local linearization to obtain a third-order Active Flux method. For Burgers’ equation ∂tu + u∂xu = 0, instead of the local linearization x →7 x − u(x)t, for example, [Roe17] suggests to use x →7 x − 1~~ ~~+utx)tu(x) ≃ x − u(x)t + t2 u (x)∂xu (x) + O(t3 ) (1) 1 CNRS, Institut de Math´ematiques de Bordeaux (IMB), UMR 5251, 351 Cours de la Lib´eration, 33405 Talence, France, [wasilij.barsukow@math.u-bordeaux.fr](wasilij.barsukow@math.u-bordeaux.fr) while [Bar21a] proposes to iterate x →7 x − u 􀀐 x − u(x)t􀀑 t ≃ x − u(x)t + t2 u (x)∂xu (x) + O(t3 ) (2) In [Bar21a], a sufficiently accurate approximate evolution operator was achieved for any hyperbolic system of conservation laws in one spatial dimension. It has been shown that it is by no means sufficient to merely iterate the linearization a few times to achieve third order of accuracy – this only works for scalar conservation laws since their characteristics are straight even in the nonlinear case (see Section 3.4 for further explanations) . An alternative, ADER-inspired approach is [HKS19] . Concerning multiple spatial dimensions, early effort ([Fan17, BHKR19]) focused on the equations of linear acoustics and the corresponding evolution operator. The exact solution for a general class of data (in particular those not differentiable) was obtained in [BK22], initially destined to study genuinely multi-dimensional (multi-d) Godunov methods. The exact solution was used to demonstrate that even the complete solution of the multidimensiona

FAQs

How does the method split the Euler equations in time evolution?

It applies additive operator splitting into an acoustic operator and a nonlinear advection operator, treating each part with a different evolution strategy.

What is used to achieve third-order accuracy for the advection operator?

The advection operator is solved at third-order accuracy using a new approximate evolution operator, rather than relying solely on repeated local linearization.

How does the method handle discontinuous solutions and maintain bounds?

It uses a blended bound preserving limiting strategy that combines a priori and a posteriori approaches to control discontinuities.

Jul 16, 2026

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