A Second-Order Unsplit Godunov Scheme for Cell-Centered MHD: the CTU-GLM scheme

We assess the validity of a single step Godunov scheme for the solution of the magneto-hydrodynamics equations in more than one dimension. The scheme is second-order accurate and the temporal discretization is based on the dimensionally unsplit Corner Transport Upwind (CTU) method of Colella. The proposed scheme employs a cell-centered representation of the primary fluid variables (including magnetic field) and conserves mass, momentum, magnetic induction and energy. A variant of the scheme, which breaks momentum and energy conservation, is also considered. Divergence errors are transported out of the domain and damped using the mixed hyperbolic/parabolic divergence cleaning technique by Dedner et al. (J. Comput. Phys., 175, 2002). The strength and accuracy of the scheme are verified by a direct comparison with the eight-wave formulation (also employing a cell-centered representation) and with the popular constrained transport method, where magnetic field components retain a staggered collocation inside the computational cell. Results obtained from two- and three-dimensional test problems indicate that the newly proposed scheme is robust, accurate and competitive with recent implementations of the constrained transport method while being considerably easier to implement in existing hydro codes.Comment: 31 Pages, 16 Figures Accepted for publication in Journal of Computational Physic

The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions

6 pagesInternational audienceWe consider hyperbolic scalar conservation laws with discontinuous flux function of the type \partial_t u + \partial_x f(x,u) = 0 \text{\;\;with\;\;} f(x,u) = f_L(u) \Char_{\R^-}(x) + f_R(u) \Char_{\R^+}(x). Here $f_{L,R}$ are compatible bell-shaped flux functions as appear in numerous applications. It was shown by Adimurthi, S. Mishra, G. D. V. Gowda ({\it J. Hyperbolic Differ. Equ. 2 (4) (2005) 783-837)} and R. Bürger, K. H. Karlsen and J. D. Towers ({\it SIAM J. Numer. Anal. 47~(3) (2009) 1684--1712}) that several notions of solution make sense, according to a choice of the so-called $(A,B)$-connection. In this note, we remark that every choice of connection $(A,B)$ corresponds to a limitation of the flux under the form $f(u)|_{x=0}\leq \bar F$, first introduced by R. M. Colombo and P. Goatin ({\it J. Differential Equations 234 (2) (2007) 654-675}). Hence we derive a very simple and cheap to compute explicit formula for the Godunov numerical flux across the interface $\{x=0\}$, for each choice of connection. This gives a simple-to-use numerical scheme governed only by the parameter $\bar F$. A numerical illustration is provided

An exactly conservative particle method for one dimensional scalar conservation laws

A particle scheme for scalar conservation laws in one space dimension is presented. Particles representing the solution are moved according to their characteristic velocities. Particle interaction is resolved locally, satisfying exact conservation of area. Shocks stay sharp and propagate at correct speeds, while rarefaction waves are created where appropriate. The method is variation diminishing, entropy decreasing, exactly conservative, and has no numerical dissipation away from shocks. Solutions, including the location of shocks, are approximated with second order accuracy. Source terms can be included. The method is compared to CLAWPACK in various examples, and found to yield a comparable or better accuracy for similar resolutions.Comment: 29 pages, 21 figure

Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly non-local model

In this paper, we propose a macroscopic model that describes the influence of a slow moving large vehicle on road traffic. The model consists of a scalar conservation law with a non-local constraint on the flux. The constraint level depends on the trajectory of the slower vehicle which is given by an ODE depending on the downstream traffic density. After proving well-posedness, we first build a finite volume scheme and prove its convergence, and then investigate numerically this model by performing a series of tests. In particular, the link with the limit local problem of [M.L. Delle Monache and P. Goatin, J. Differ. Equ. 257(11), 4015-4029 (2014)] is explored numerically