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

Highlights: • Two new families of high-order well-balanced finite difference schemes for 1d systems of balance laws are introduced. • While the methods of the first family preserve every stationary solution, those of the second one preserve a given set of them. • The accuracy, well-balancedness, and conservation properties of the methods are analyzed. • The difficulties posed by a singular source term are addressed. • Extensive numerical experiments which validate the theoretical properties are shown. In this paper, high order well-balanced finite difference weighted essentially non-oscillatory methods to solve general systems of balance laws are presented. Two different families are introduced: while the methods in the first one preserve every stationary solution, those in the second family only preserve a given set of stationary solutions that depend on some parameters. The accuracy, well-balancedness, and conservation properties of the methods are discussed, as well as their application to systems with singular source terms. The strategy is applied to derive third and fifth order well-balanced methods for a linear scalar balance law, Burgers' equation with a nonlinear source term, and for the shallow water model. In particular, numerical methods that preserve every stationary solution or only water at rest equilibria are derived for the latter.