Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 2652367, 15 pages

https://doi.org/10.1155/2018/2652367

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

Correspondence should be addressed to Hongwei Yang

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.

#### 1. Introduction

In this paper, the simple finite volume WENO (weighted essentially nonoscillatory) scheme with Lax-Wendroff-type time discretization is proposed to numerically solve the system of shallow water equations with bottom topography influence in source terms.

It has been an important work to search for the solutions of nonlinear differential equations due to their rich mathematical structures and features [1–4] as well as important applications in fluid dynamics and plasma physics [5–9]. The solutions of differential equations can be simply divided into two kinds, exact solutions and numerical solutions. For the exact solutions, a great number of methods have been introduced, like Painlevé test, Darboux Transformation, bilinear method, symmetry method [10], and so on [11–19], while for the forced nonlinear differential equations, such as shallow water equations with topography influence in source terms, it is very difficult to get the exact solutions and we have to derive the numerical solutions and analyze the topography forced effect.

WENO is a procedure of spatial discretization for partial differential equation (PDE); in other words, it is a numerical method to discretize the derivative terms in space. The WENO scheme is an important high order accuracy numerical method, especially to simulate strong shocks and contact discontinuous and complicated smooth solutions, as they can keep high order and nonoscillatory characteristic for both continuous and discontinuous solutions. WENO scheme is popularly used in many communities [20–23] in recent years; it has attracted much attention in CFD (computational fluid dynamics); especially for shallow water flows, finite volume WENO scheme made important contributions as more flexibility of mesh shape. Xing and Shu [24] introduced a finite volume WENO wave propagation scheme on rectangular mesh. Lu et al. [25] investigate the performance of finite volume WENO scheme for the system of shallow water equations on unstructured triangle meshes; simulation of a tidal bore on Qiantang river is performed.

In recent years, WENO scheme has been generalized for hyperbolic conservation laws [22, 26, 27] after the first WENO scheme was originally derived in [28] for the third-order finite volume frame based on ENO type schemes [29, 30], such as CWENO and hybrid WENO [31] arising from various applications. In 2016, a successful type of WENO [32] was proposed to approximate hyperbolic conservation laws. The new WENO reconstruction is a convex combination of two linear polynomials with a fourth-degree polynomial using the same five-point big stencil as in the classic WENO fashion; the linear weights are constants, which are positive, and their summation equals one so they are more simple and easily extended to multidimensions in engineering applications. We call the new type of WENO as simple WENO (SWENO).

For time-dependent problems, there are mainly two different ways to approximate derivative with respect to time [33–35]. One common approach is well known ODE solver, such as Euler’ method, Runge-Kutta method, Adams’ multilevel method, and Radau scheme. These approaches have been popularly used for the advantages of good stability and simplicity in idea and codes. The disadvantage is that the highest accuracy order for total variation diminishing (TVD) Runge-Kutta method is fourth order. Another way is via the Lax-Wendroff-type procedure; the idea is converting all the time derivatives into spatial derivatives using PDE and Taylor expansion with respect to time, then discretizing the spatial derivatives using numerical methods. The advantages are smaller stencil and frequent use of the original PDE. The disadvantages are complicated in formulation and coding. Both approaches can produce any order accuracy in time theoretically. The finite volume Lax-Wendroff time discretization formula is a little simpler than that of the finite difference frame. Qiu and Shu [34] derived a scheme consisting of Lax-Wendroff-type time discretization and the finite difference WENO scheme in compressible gas dynamics; the scheme has important contributions to reducing CPU cost and maintaining the nonoscillatory characteristic. Lu and Qiu [36] also developed Lax-Wendroff with finite difference WENO scheme for shallow water equations; similar results were obtained.

In this paper, we developed the fifth-order SWENO finite volume scheme with the third-order Lax-Wendroff-type time discretization to simulate shallow water flows. In the scheme we follow the ideas in [32, 37] about simple finite volume WENO scheme and the algebraic prebalance method for the flux and the source terms [36, 38] in shallow water equations. An outline of the paper is as follows: in Section 2 we describe details of the discretization with finite volume SWENO scheme and Lax-Wendroff-type time discretization for shallow water equations. In Section 3, a range of 1D and 2D numerical results show that the scheme has accuracy, resolution, efficiency, and robustness. Finally, Section 4 contains conclusions.

#### 2. Description of Numerical Model

##### 2.1. The Shallow Water Equations

The 2D system of shallow water equations with conservative form iswith conservational vector fluxesand source termswhere is the total water depth; and are the depth-averaged water velocities in the - and -directions, respectively; is the time; is the gravity constant; is the topography function above; and are derivatives with respect to and ; they are used to describe the bed slopes stresses. Friction on the bottom, wind stress, and Coriolis term could also be considered as parts of source terms.

We rewrite (1) in quasilinear form aswhere The eigenvalues of Jacobian matrixes and areThe normalized right and left eigenvectors of Jacobian matrixes and are as follows:We can easily check that , , which is an important characteristic for hyperbolic conservation law. Neglecting variables in -direction, we have the 1D conservative unsteady shallow water equations: Similar way is that we can get Jacobian matrix and other information for 1D case.

##### 2.2. The Fifth-Order Finite Volume SWENO Schemes

The fifth-order finite volume SWENO scheme will be used to numerically solve the spatial derivatives of shallow water equations [32, 37]. In this section, the basic procedure of SWENO will be described as short overview for the following 1D scalar hyperbolic conservation law:

Denote as the th cell, a control volume for finite volume method, centered on ; we assume that the grid points are uniform with . For a finite volume scheme, we evolve the proximate average value of at mesh cell in time. Using integral average, the conservation law (10) is rewritten into is approximated by numerical flux . The popular Lax-Friedrich flux is used for its simpleness and monotone, which is given by where ; are approximations to of left side and right side at considering the discontinuous solution. We call this process reconstruction; here the reconstructions of interface values depend on the cell averages by the fifth-order SWENO reconstruction procedure.

Only will be introduced in detail as follows. For the fifth-order SWENO scheme, three approximated point values of should be constructed. Choose the first big stencil ; a fourth-degree polynomial can be obtained by requiring the same mean value on the five cells. The first approximated point value for isChoose the second stencil ; another linear polynomial can be obtained with the same idea. The second approximation isChoose the third stencil ; a linear polynomial can be obtained and the third approximation is

Convex combination coefficients are the nonlinear weights. Before that we have to compute the smoothness indicators of the three above polynomials first. For the fifth-order SWENO reconstructions, the convex combination of all these three-point values is The nonlinear weights are defined by and here is meaningless, just to avoid the denominator to be zero, so is a positive constant and typically , are the linear weights, and “smoothness indicators” are used to measure the smoothness of the three polynomials in cell . The smaller the value of smooth indicator is, the smoother the polynomial is. We use the same smooth indicator formula for -degree polynomial in [39]The explicit expression of the three smoothness indicators is given byFor the fifth-order SWENO scheme on uniform mesh, the important idea is that the linear weights are your chosen positive constants; the only condition is that the summation is one. So the procedure of linear weights in classical WENO is avoided, and the negative weights are prevented. In this paper, we use the positive linear weights as

On uniform mesh, the reconstruction of is mirror symmetry on big stencil to that of . For shallow water equations, the reconstruction is operated in local eigenspace for more robust, so the local characteristic decomposition should be used, in which properties of eigenspace and Roe’s averages are needed.

For 2D finite volume case, we cannot simply proceed in a dimension-by-dimension fashion as in finite difference method. The spatial surface integration on rectangular cell can be converted to line integration by Gauss theorem. In order to get the fifth order, we can use three-point Gaussian quadrature for line integration of every rectangular cell like in Figure 1. For example, to calculate numerical flux in -direction, three Gaussian point values on every vertical line should be reconstructed from cell average values in -direction first, then left and right Gaussian point values should be reconstructed by the SWENO scheme, then the numerical fluxes are calculated, and last the cell average value can be approximated on next time level.