scholarly journals Stationarity Preservation Properties of the Active Flux Scheme on Cartesian Grids

2020 ◽  
Author(s):  
Wasilij Barsukow
Keyword(s):  
Evolution Operator ◽  
Hyperbolic Systems ◽  
Finite Volume Scheme ◽  
Stationary States ◽  
Flux Method ◽  
Spatial Dimensions ◽  
Systems Of Conservation Laws ◽  
The One ◽  
Active Flux ◽  
Additional Point

Abstract Hyperbolic systems of conservation laws in multiple spatial dimensions display features absent in the one-dimensional case, such as involutions and non-trivial stationary states. These features need to be captured by numerical methods without excessive grid refinement. The active flux method is an extension of the finite volume scheme with additional point values distributed along the cell boundary. For the equations of linear acoustics, an exact evolution operator can be used for the update of these point values. It incorporates all multi-dimensional information. The active flux method is stationarity preserving, i.e., it discretizes all the stationary states of the PDE. This paper demonstrates the experimental evidence for the discrete stationary states of the active flux method and shows the evolution of setups towards a discrete stationary state.

scholarly journals The Active Flux Scheme for Nonlinear Problems

2020 ◽  
Vol 86 (1) ◽  
Author(s):  
Wasilij Barsukow
Keyword(s):  
Hyperbolic Systems ◽  
Nonlinear Problems ◽  
Cell Boundary ◽  
Finite Volume Scheme ◽  
Evolution Operators ◽  
Third Order ◽  
Order Of Accuracy ◽  
Wave Speeds ◽  
Spatial Dimensions ◽  
Active Flux

AbstractThe Active Flux scheme is a finite volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver. Instead, given a reconstruction, the initial value problem at the location of the point value is solved. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. Whereas for linear problems an exact evolution operator is available, for nonlinear problems one needs to resort to approximate evolution operators. This paper presents such approximate operators for nonlinear hyperbolic systems in one dimension and nonlinear scalar equations in multiple spatial dimensions. They are obtained by estimating the wave speeds to sufficient order of accuracy. Additionally, an entropy fix is introduced and a new limiting strategy is proposed. The abilities of the scheme are assessed on a variety of smooth and discontinuous setups.

A REDUCED STABILITY CONDITION FOR NONLINEAR RELAXATION TO CONSERVATION LAWS

2004 ◽  
Vol 01 (01) ◽  
pp. 149-170 ◽  
Author(s):  
FRANÇOIS BOUCHUT
Keyword(s):  
Conservation Laws ◽  
Stability Condition ◽  
Hyperbolic Systems ◽  
Strong Stability ◽  
Riemann Solvers ◽  
Systems Of Conservation Laws ◽  
Leading Term ◽  
Relaxation Systems ◽  
The One ◽  
Approximate Riemann Solvers

We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore and Liu, namely the existence of an entropy extension. We propose here a new stability condition, the reduced stability condition, which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the Chapman–Enskog expansion. This reduced stability condition has the advantage of involving only the submanifold of equilibria, or maxwellians, so that it is much easier to check than the entropy extension condition. Our condition generalizes the one introduced by the author in the case of kinetic, i.e. diagonal semilinear relaxation. We provide an adapted stability analysis in the context of approximate Riemann solvers obtained via relaxation systems.

scholarly journals On the uniqueness of solutions to hyperbolic systems of conservation laws

2021 ◽  
Vol 291 ◽  
pp. 110-153
Author(s):  
Shyam Sundar Ghoshal ◽  
Animesh Jana ◽  
Konstantinos Koumatos
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Systems ◽  
Uniqueness Of Solutions ◽  
Systems Of Conservation Laws

The Cartesian Grid Active Flux Method: Linear stability and bound preserving limiting

2021 ◽  
Vol 393 ◽  
pp. 125501
Author(s):  
Erik Chudzik ◽  
Christiane Helzel ◽  
David Kerkmann
Keyword(s):  
Linear Stability ◽  
Cartesian Grid ◽  
Flux Method ◽  
Active Flux

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

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