Active Flux Schemes for Systems

Eymann, Timothy A.; Roe, Philip L.

doi:10.2514/6.2011-3840

Cited by 24 publications

(19 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A case where the flow is irrotational was presented in. 5 At that time it was thought that the rotational case would need to be handled by a different formula, perhaps with extra terms, but there is however, another form of the Poisson formula that can be seen to apply to all variables. Manipulation allows (6) to be rewritten as…”

Section: B the Poisson Solution To The Initial Value Problemmentioning

confidence: 98%

See 1 more Smart Citation

Investigations of a New Scheme for Wave Propagation

Fan

Roe

2015

22nd AIAA Computational Fluid Dynamics Conference

View full text Add to dashboard Cite

We conduct a comparison of the Active Flux method proposed by Eymann and Roe versus Discontinuous Galerkin with linear reconstruction (DG1). We find the Active Flux method capable of matching the accuracy of DG1 with a mesh spacing about three times greater, and capable of time steps about 2.5 times longer. On a given mesh, the Active Flux method can compute flows that should display circular symmetry with dramatically less scatter.

show abstract

Section: B the Poisson Solution To The Initial Value Problemmentioning

confidence: 98%

“…5 The method is capable of solving wave equations, such as those governing acoustic waves, electromagnetic phenomena, or elastic waves, on arbitrary unstructured grids. For linear problems it can be implemented to any order, and is exact on any grid whenever the initial data is a global polynomial of the same degree as the reconstruction.…”

Section: Introductionmentioning

confidence: 99%

Investigations of a New Scheme for Wave Propagation

Fan

Roe

2015

22nd AIAA Computational Fluid Dynamics Conference

View full text Add to dashboard Cite

show abstract

“…Hence, the algorithm presented is equally applicable to steady and unsteady computations. The active flux scheme is already available individually for the advective term [1] and the diffusive term [6]. In some methods, a third-order advection-diffusion scheme can be easily constructed by adding the advection and diffusion schemes.…”

Section: Advection-diffusion Equationmentioning

confidence: 99%

“…[1,2], built upon Scheme V of Van Leer [3], as a viable alternative to high-order methods. Active flux schemes are finitevolume-based compact high-order schemes.…”

Section: Introductionmentioning

confidence: 99%

Active Flux for Advection Diffusion

Nishikawa

2015

22nd AIAA Computational Fluid Dynamics Conference

View full text Add to dashboard Cite

In this paper, we extend the third-order active-flux diffusion scheme introduced in Ref.[AIAA Paper 2014-2092] to advection diffusion problems. It is shown that a third-order active-flux advection-diffusion scheme can be constructed by adding the advective term as a source term to the diffusion scheme. The solution gradient, which is computed simultaneously to thirdorder accuracy by the diffusion scheme, is used to express the advective term as a scalar source term. To solve the residual equations efficiently, Newton's method is employed rather than explicit pseudo-time iterations, which requires a large number of residual evaluations. For unsteady computations, it leads to a third-order implicit time-stepping scheme with Newton's method. Numerical experiments show that the resulting advection-diffusion scheme achieves third-order accurate and the Newton solver converges very rapidly for both steady and unsteady problems.

show abstract

“…Note that it is the piecewise quadratic reconstruction which limits the method in its current form to third order accuracy. Eymann and Roe [3,4] used characteristic theory in order to compute q(x i+ 1 2 , t n+ 1…”

mentioning

confidence: 99%

A New ADER Method Inspired by the Active Flux Method

2019

View full text Add to dashboard Cite

We study numerical methods that are inspired by the active flux method of Eymann and Roe and present several new results for one and two-dimensional hyperbolic problems. For one-dimensional linear problems we show that the unlimited active flux method can be interpreted as an ADER method. This interpretation motivates the construction of new third order accurate methods for nonlinear hyperbolic conservation laws. In the two-dimensional case, equivalent methods are only obtained for scalar linear problems. For two-dimensional linear systems the methods are no longer equivalent. For the two-dimensional acoustic equations we compare the accuracy of the two resulting approaches. While commonly used methods for hyperbolic problems are based on discontinuous reconstructions, the active flux method uses a continuous, piecewise quadratic reconstruction. For nonlinear problems we identify a situation in which the continuous reconstruction leads to an unstable approximation. We propose a limiting strategy which overcomes this problem. Our limited version of the active flux method uses the same local stencil as the original method. Keywords Finite volume methods • Hyperbolic conservation laws • Active flux method • ADER method • High-order methods Mathematics Subject Classification 65M08 • 65M25 Recently, Eymann, Roe and coauthors [2-4,12,14] introduced a new numerical method for hyperbolic conservation laws, which they called the active flux method. For sufficiently smooth linear problems the method is third order accurate [4]. In [12], third order accurate This work was supported by the DFG through HE 4858/4-1.

show abstract

Active Flux Schemes for Systems

Cited by 24 publications

References 19 publications

Investigations of a New Scheme for Wave Propagation

Investigations of a New Scheme for Wave Propagation

Active Flux for Advection Diffusion

A New ADER Method Inspired by the Active Flux Method

Contact Info

Product

Resources

About