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In this paper, high order well-balanced finite difference weightedessentially non-oscillatory methods to solve general systems of balance lawsare presented. Two different families are introduced: while the methods in thefirst one preserve every stationary solution, those in the second family onlypreserve a given set of stationary solutions that depend on some parameters.The accuracy, well-balancedness, and conservation properties of the methods arediscussed, as well as their application to systems with singular source terms.The strategy is applied to derive third and fifth order well-balanced methodsfor a linear scalar balance law, Burgers' equation with a nonlinear sourceterm, and for the shallow water model. In particular, numerical methods thatpreserve every stationary solution or only water at rest equilibria are derivedfor the latter.

Well-balanced high-order finite difference methods for systems of balance laws