Active Flux Schemes

21st AIAA Computational Fluid Dynamics Conference

2013

We build on active flux (AF) schemes and extend the method to the two-dimensional linear advection, linear acoustics, and linearized Euler equations. The active flux method independently updates edge and centroid values. Because the interface fluxes are calculated from independently updated quantities, we say that they are actively updated, as opposed to a more traditional finite-volume scheme where the fluxes are updated passively from the conserved values. The one-dimensional active flux method is reformulated using Lagrange basis functions and the basic features of the method are reviewed. The two-dimensional formulation also uses standard basis functions, but includes a bubble function to maintain conservation. A novel approach for solving the linear wave system is presented that uses spherical means to compute the edge updates for the flux calculation. We demonstrate that the AF method is third-order accurate for advection, acoustics, and the linearized Euler equations.

Section: Accuracymentioning

confidence: 99%

Multidimensional Active Flux Schemes

Eymann

21st AIAA Computational Fluid Dynamics Conference

2013

“…The active flux method is a third-order finite volume method developed by Eymann and Roe [5][6][7][8], building on the work of van Leer [9]. We briefly review the method for a fist-order conservation law of the form r(u) ≡ ∂u ∂t…”

Section: The Active Flux Methodsmentioning

confidence: 99%

Continuous adjoint based error estimation and r-refinement for the active-flux method

Ding

Fidkowski

54th AIAA Aerospace Sciences Meeting

2016

This paper explores the possibility of using continuous adjoints to estimate errors in scalar outputs and to drive mesh adaptation for the active-flux method. Compared to a discrete adjoint approach, the continuous adjoint offers a few attractive advantages for the active-flux method, such as a naturally fully-discrete, explicit implementation that parallels the simplicity of the primal system evolution and error estimates obtained from integrals over the space-time control volumes. The latter point is important because the active flux method does not possess a variational formulation. In addition, we present novel unsteady adaptation mechanics that rely on dynamically moving mesh nodes, i.e. r-refinement, in order to minimize the output error.

“…Together with my student Timothy Eymann, we derived a family of secondorder schemes of this form for one-dimensional linear advection, and came to realize [9] that the optimal, third-order, member of the family was in fact the scheme derived in 1977 by van Leer as part of his "new approach to numerical advection" [40] and denoted by him as Scheme V. An appreciation of van Leer's paper, with some additional one-dimensional schemes, has been presented by Rider [31]. Several authors since van Leer have rediscovered this advection scheme, apparently independently.…”

Section: What Else Could There Be?mentioning

confidence: 99%

“…The first is to update the fluxes by a second-order upwind scheme, either by (5), or by (6), (7). The second component is a central leapfrog scheme (9). This interpretation will be the key to multidimensional generalizations.…”

Section: Advection In One Dimensionmentioning

confidence: 99%

Is Discontinuous Reconstruction Really a Good Idea?

2017

J Sci Comput

Self Cite

It has been almost automatically assumed for a quarter century that the numerical solution of hyperbolic conservation laws is best accomplished by making a reconstruction of the initial data that is only piecewise continuous. The effect of the discontinuities is taken into account by means of Riemann solvers. This strategy has enjoyed great practical success but introduces only one-dimensional physics as a guide to the discretization of multidimensional problems. This article points out some of the resulting defects and proposes an alternative viewpoint. The chief novelty of the new "Active Flux" method, apart from the elimination of discontinuities, is the division into advective and acoustic disturbances, with acoustics being handled by exploiting classical solutions to the scalar wave equation. Results for a standard test problem for the compressible Euler equations indicate that an order of magnitude increase in cost effectiveness may be possible by following this approach, compared to the traditional one.