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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Highlights • Coupled Roe's scheme for SW-Exner equations in cross-section averaged erosive flows. • Explicit expressions for eigenvalues, eigenvectors, wave and source strengths. • A new parameter c b related to the bed change appears in the coupled Jacobian matrix. • c b depends on the erosion-deposition mechanism selected to update the cross-section. • Analytical expression for the equilibrium bed slope in variable width rectangular channels.

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