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In the context of preserving stationary states, e.g. lake at rest and movingequilibria, a new formulation of the shallow water system, called FluxGlobalization has been introduced by Cheng et al. (2019). This approachconsists in including the integral of the source term in the global flux andreconstructing the new global flux rather than the conservative variables. Theresulting scheme is able to preserve a large family of smooth and discontinuoussteady state moving equilibria. In this work, we focus on an arbitrary highorder WENO Finite Volume (FV) generalization of the global flux approach. Themost delicate aspect of the algorithm is the appropriate definition of thesource flux (integral of the source term) and the quadrature strategy used tomatch it with the WENO reconstruction of the hyperbolic flux. When thisconstruction is correctly done, one can show that the resulting WENO FV schemeadmits exact discrete steady states characterized by constant global fluxes. Wealso show that, by an appropriate quadrature strategy for the source, we canembed exactly some particular steady states, e.g. the lake at rest for theshallow water equations. It can be shown that an exact approximation of globalfluxes leads to a scheme with better convergence properties and improvedsolutions. The novel method has been tested and validated on classical cases:subcritical, supercritical and transcritical flows.

Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation