The Flux Reconstruction Method with Laxxe2x80x93Wendroff Type Temporal Discretization for Hyperbolic Conservation Laws In this paper, we develop a Laxxe2x80x93Wendroff type time discretization method for high order Flux Reconstruction scheme to solve hyperbolic conservation laws. Through Cauchyxe2x80x93Kowalewski procedure, the resulting Laxxe2x80x93Wendroff Flux Reconstruction (LWFR) scheme is an alternative spatialxe2x80x93temporal coupling method to the popular Rungexe2x80x93Kutta Flux Reconstruction (RKFR) scheme. LWFR is a onestep explicit high order discontinuous finite element method and its discretization procedure is more compact and effective for certain problems than that of RKFR. Furthermore, aiming at accurate simulation of discontinuity, we propose a robust local artificial viscosity formulation of LWFR for the first time. A collection of successful numerical experiments show that LWFR can give essentially nonoscillatory and sharp solutions for discontinuity, and maintain designed order of accuracy for smooth regions, both in onedimensional and twodimensional Euler equations. In conclusion, LWFR scheme is cost effective and accuracypreserving for certain problems and can be an attractive candidate to solve hyperbolic conservation laws.

WellBalanced HighOrder Finite Volume Methods for Systems of Balance Laws In some previous works, the authors have introduced a strategy to develop wellbalanced highorder numerical methods for nonconservative hyperbolic systems in the framework of pathconservative numerical methods. The key ingredient of these methods is a wellbalanced reconstruction operator, i.e. an operator that preserves the stationary solutions in some sense. A strategy has been also introduced to modify any standard reconstruction operator like MUSCL, ENO, CWENO, etc. in order to be wellbalanced. In this article, the specific case of 1d systems of balance laws is addressed and difficulties are gradually introduced: the methods are presented in the simpler case in which the source term does not involve Dirac masses. Next, systems whose source term involves the derivative of discontinuous functions are considered. In this case, the notion of weak solution is discussed and the Generalized Hydrostatic Reconstruction technique is used for the treatment of singular source terms. A technique to preserve the wellbalancedness of the methods in the presence of numerical integration is introduced. The strategy is applied to derive first, second and third order wellbalanced methods for Burgersxe2x80x99 equation with a nonlinear source term and for the Euler equations with gravity.