We present a finite volume method for the numerical solution of the sediment transport equations in one and two space dimensions. The numerical fluxes are reconstructed using a modified Roe scheme that incorporates, in its reconstruction, the sign of the Jacobian matrix in the sediment transport system. A well-balanced discretization is used for the treatment of source terms. The method is well balanced, nonoscillatory, and suitable for both structured and unstructured triangular meshes. An adaptive procedure is also considered for the two-dimensional problems to update the bed-load accounting for the interaction between the bed-load and the water flow. The proposed method is applied to several sediment transport problems in one and two space dimensions. The numerical results demonstrate high resolution of the proposed finite volume method and confirm its capability to provide accurate simulations for sediment transport problems under flow regimes with strong shocks.

In this paper, we establish some fixed point results for α-λ-contractions in the class of quasi b-metric spaces. To illustrate the obtained results, we provide some examples and an application on a solution of an integral equation. We also study the stability of Ulam-Hyers and well-posedness of a fixed point problem. Our obtained results give an answer to an open problem of Kutbi and Sintunavarat (Abstr. Appl. Anal. 2014:268230, 2014 ).