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Mathematical Problems in Engineering
Volume 2018, Article ID 2652367, 15 pages
https://doi.org/10.1155/2018/2652367
Research Article

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

1School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
3Key Laboratory of Meteorological Disaster of Ministry of Education, Nanjing University of Information Science and Technology, Nanjing 210044, China

Correspondence should be addressed to Hongwei Yang; moc.361@9791gnaywh

Received 1 November 2017; Accepted 24 January 2018; Published 22 February 2018

Academic Editor: Marco Pizzarelli

Copyright © 2018 Changna Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.