Active Flux is a recently developed numerical method for hyperbolic conservation laws. Its classical degrees of freedom are cell averages and point values at interfaces. These latter shared between adjacent cells, leading to globally continuous reconstruction. The update the includes upwinding, but without solving Riemann Problem. average requires flux interface, which can be immediately obtained using values. This paper explores different extensions arbitrarily high order accuracy, while...

ABSTRACT Numerical methods for hyperbolic PDEs require stabilization. For linear acoustics, divergence‐free vector fields should remain stationary, but classical Finite Difference add incompatible diffusion that dramatically restricts the set of discrete stationary states numerical method. Compatible vanish on states, e.g., there be a gradient divergence. Some Element allow natural embedding this grad‐div structure, SUPG method or OSS. We prove here particular discretization associated with...

The Active Flux method can be seen as an extended finite volume method. degrees of freedom this are cell averages, in methods, and addition shared point values at the interfaces, giving rise to a globally continuous reconstruction. Its classical version was introduced one-stage fully discrete, third-order Recently, semi-discrete presented with various extensions arbitrarily high order one space dimension. In paper we extend on two-dimensional Cartesian grids order, by including moments...

Finite volume schemes often have difficulties to resolve the low Mach number (incompressible) limit of Euler equations. Incompressibility is only non-trivial in multiple spatial dimensions. Low fixes, however generally are applied one-dimensional method and then used a dimensionally split way. This reduces its stability. Here, it suggested keep as is, extend dimensions particular, all-speed strategy found lead much more stable numerical methods. Apart from conceptually pleasing property...

The acoustic equations derived as a linearization of the Euler are valuable system for studies multi-dimensional solutions. Additionally they possess low Mach number limit analogous to that equations. Aiming at understanding behaviour Godunov scheme in this limit, first exact solution corresponding Cauchy problem three spatial dimensions is derived. appearance logarithmic singularities 4-quadrant Riemann Problem two discussed. formulae then used obtain multidimensional finite volume...

There is a qualitative difference between one-dimensional and multi-dimensional solutions to the Euler equations: new features that arise are vorticity nontrivial incompressible (low Mach number) limit. They present challenges finite volume methods. It seems an important step in this direction first study for acoustic equations. exists analogue of low number limit system its stationary. shown scheme possesses stationary discrete (vorticity preserving) also has states discretizations all...

Abstract The Active Flux scheme is a finite volume with additional point values distributed along the cell boundary. It third order accurate and does not require Riemann solver. Instead, given reconstruction, initial value problem at location of solved. intercell flux then obtained from evolved boundary by quadrature. Whereas for linear problems an exact evolution operator available, nonlinear one needs to resort approximate operators. This paper presents such operators hyperbolic systems in...

In order to perform simulations of low Mach number flow in presence gravity the technique from [23] is found insufficient as it unable cope with a hydrostatic equilibrium.Instead, new modification diffusion matrix context Roe-type schemes suggested.We show that without able resolve incompressible limit, and does not violate conditions equilibrium when present.These properties are verified by performing formal asymptotic analysis scheme.Furthermore, we study its von Neumann stability subject...

Recently Pasetto et al. have proposed a new method to derive convection theory appropriate for the implementation in stellar evolution codes. Their approach is based on simple physical picture of spherical bubbles moving within potential flow dynamically unstable regions, and detailed computation bubble dynamics. Based this approach, authors which claimed be parameter-free, non-local time-dependent. This very strong claim, as such holy grail physics. Unfortunately, we identified several...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 19 June 2020Accepted: 03 August 2021Published online: 14 December 2021Keywordsfinite volume methods, flux, source terms, balance laws, well-balanced gravityAMS Subject Headings35L65, 35L45, 65M08Publication DataISSN (print): 1064-8275ISSN (online): 1095-7197Publisher: Society for Industrial and Applied MathematicsCODEN: sjoce3

High-order Godunov methods for gas dynamics have become a standard tool simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct pair Riemann states at cell interfaces and solver that computes interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so many schemes in principle built from same numerical framework. this work, we use our fully compressible Seven-League Hydro...

High-order Godunov methods for gas dynamics have become a standard tool simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct pair Riemann states at cell interfaces and solver that computes interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so many schemes in principle built from same numerical framework. Because simulations often test out limits what feasible with...

The active flux (AF) method is a compact high-order finite volume that evolves cell averages and point values at interfaces independently. Within the of lines framework, value can be updated based on Jacobian splitting (JS), incorporating upwind idea. However, such JS-based AF methods encounter transonic issues for nonlinear problems due to inaccurate direction estimation. This paper proposes use vector update, offering natural uniform remedy issue. To improve robustness, this also develops...