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.

PositivityPreserving Finite Difference WENO Scheme for TenMoment Equations with Source Term We develop a positivitypreserving finite difference WENO scheme for the TenMoment equations with body forces acting as a source in the momentum and energy equations. A positive forward Euler scheme under a CFL condition is first constructed which is combined with an operator splitting approach together with an integrating factor, strong stability preserving Rungexe2x80x93Kutta scheme. The positivity of the forward Euler scheme is obtained under a CFL condition by using a scaling type limiter, while the solution of the source operator is performed exactly and is positive without any restriction on the time step. The proposed method can be used with any WENO reconstruction scheme and we demonstrate it with fifth order accurate WENOJS, WENOZ and WENOAO schemes. An adaptive CFL strategy is developed which can be more efficient than the use of reduced CFL for positivity preservation. Numerical results show that high order accuracy and positivity preservation are achieved on a range of test problems.