High Order StillWater and MovingWater Equilibria Preserving Discontinuous Galerkin Methods for the Ripa Model Shallow water equations with horizontal temperature gradients, also known as the Ripa system, are used to model flows when the temperature fluctuations play an important role. These equations admit steady state solutions where the fluxes and source terms balance each other. We present wellbalanced discontinuous Galerkin methods for the Ripa model which can preserve the stillwater or the general movingwater equilibria. The key ideas are the recovery of wellbalanced states, separation of the solution into the equilibrium and fluctuation components, and appropriate approximations of the numerical fluxes and source terms. The same framework is also extended to design wellbalanced methods for the constant height and isobaric steady state solutions of the Ripa model. Numerical examples are presented to verify the wellbalanced property, high order accuracy, and good resolution for both smooth and discontinuous solutions.

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.