A second-order flux globalization based well-balanced (WB) path-conservative central-upwind (PCCU) scheme for the shallow water flows in rectangular cross-section channels has been recently proposed in Chen et al. ESAIM Math. Model. Numer. Anal. 57 (2023) 1087–1110. Here, we extend this method to a more general case of arbitrary cross-section channels and to a higher (fifth) order of accuracy. One of the challenges in extending the numerical methods for the arbitrary cross-section channel system is related to converting the equilibrium variables (need to evaluate the global fluxes in a well-balanced manner) to the conservative ones. Unlike the case of rectangular cross-section channels, this conversion process requires the use of carefully designed nonlinear equation solver. Challenges in extending the scheme to higher-order are mainly related to the development of higher-order WB and non-oscillatory reconstructions (interpolations). The introduced WB schemes are capable of exactly preserving a wide variety of steady-state solutions including both the still- and moving-water ones. The proposed schemes are tested on a number of numerical examples with both continuous and discontinuous channel breadth.
Kurganov et al. (Thu,) studied this question.
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