Mathematical Problems in Engineering

Volume 2015, Article ID 379281, 10 pages

http://dx.doi.org/10.1155/2015/379281

## Convergence Improved Lax-Friedrichs Scheme Based Numerical Schemes and Their Applications in Solving the One-Layer and Two-Layer Shallow-Water Equations

^{1}State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China^{2}Hydrology Bureau, Yangtze River Water Resource Commission, Wuhan 430010, China^{3}Yangtze River Scientific Research Institute, Wuhan 430015, China

Received 13 August 2015; Accepted 22 October 2015

Academic Editor: Maurizio Brocchini

Copyright © 2015 Xinhua 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

The first-order Lax-Friedrichs (LF) scheme is commonly used in conjunction with other schemes to achieve monotone and stable properties with lower numerical diffusion. Nevertheless, the LF scheme and the schemes devised based on it, for example, the first-order centered (FORCE) and the slope-limited centered (SLIC) schemes, cannot achieve a time-step-independence solution due to the excessive numerical diffusion at a small time step. In this work, two time-step-convergence improved schemes, the C-FORCE and C-SLIC schemes, are proposed to resolve this problem. The performance of the proposed schemes is validated in solving the one-layer and two-layer shallow-water equations, verifying their capabilities in attaining time-step-independence solutions and showing robustness of them in resolving discontinuities with high-resolution.

#### 1. Introduction

The Lax-Friedrichs (LF) scheme, also called the Lax method [1], is a classical explicit three-point scheme in solving partial differential equations in, for example, aerodynamics, hydrodynamics, and magnetohydrodynamics [2–4]. This scheme has many advantages, for example, being stable and monotone up to Courant-Friedrichs-Lewy (CFL) number approaching unity in solving the linear convection problem [4] and being computationally efficient and simple to implement (e.g., compared with a Godunov-type scheme). Due to the excellent monotone and stable properties, the LF scheme, which is of the first-order accuracy in space, has been commonly adopted for a combination use with some higher-order schemes to predict monotone and stable solutions with lower numerical diffusion [5]. For instance, Toro and Billett [6] derived the first-order centered (FORCE) scheme, by combining the LF scheme with the two-step, second-order Lax-Wendroff scheme [7]. When solving the linear convection problem, compared with the LF scheme, the numerical viscosity of the FORCE scheme is reduced by half [4]. The FORCE scheme and its second-order (both in time and in space) extension, the slope-limited centered scheme (SLIC) [8], have been used and verified to perform well in solving the one-layer shallow-water equations (1LSWEs), the Euler equations of compressible gas dynamics, and the magnetohydrodynamic equations [8, 9].

Nevertheless, the LF scheme and the schemes devised based on it (e.g., the FORCE and SLIC schemes) have a common shortcoming, which has raised less attention so far. We demonstrate this shortcoming by using the LF scheme to solve the following linear convection problem in the time-space domain :Here, is the wave propagation speed and, without loss of generality, is assumed to be positive. An explicit finite-volume discretization for (1a) giveswhere the superscripts “” and “” denote the values at the old and new time steps, respectively; and denote the time step and space interval, respectively; denotes the numerical flux and, for the LF scheme, it reads [1]A truncation error analysis shows that the LF scheme in solving (1a), (1b), and (1c), with the discretizations given in (2) and (3), leads to a numerical viscosity of [4]where is the CFL number and, for the linear convection problem,In Figure 1, is plotted against . For clear view purpose, the relation in the region is plotted on a semilog scale (see Figure 1(a)). For convenience, in Figure 1, is assumed so that (see (5)). As seen in Figure 1, increases as decreases. When , , suggesting that the numerical diffusion is absolutely dominant over the physical convection; this is, of course, unrealistic. In numerical practice, just as usually done via implementing a grid-independence study to choose an appropriate spatial resolution (can be done for the LF scheme as when , ), a -independence study is required to choose a reasonable time step. However, due to the rapidly increased numerical diffusion when decreases, the solution computed by using the LF scheme does not converge and thus a -independence study cannot be implemented. Failing to achieve a -independence solution introduces uncertainty in a numerical solution, and the time step can only be determined empirically. Note that, in some situations, for instance, when rapid wet-dry transitions exist (e.g., for simulations in the nearshore regions), or when some other constraints on the time step (e.g., those due to surface tension and fluid viscosity effects [10]) dominate the CFL condition, one may have to use a rather small time step compared with that determined by the CFL condition. In these situations, the LF-based schemes are inappropriate due to the excessive numerical diffusion at a small value of .