Advances in Mathematical Physics

Volume 2018, Article ID 3469534, 10 pages

https://doi.org/10.1155/2018/3469534

## CIP Method of Characteristics for the Solution of Tide Wave Equations

^{1}Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, China^{2}School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Correspondence should be addressed to Kai Fu; nc.ude.cuo@ufk

Received 2 February 2018; Accepted 26 May 2018; Published 2 July 2018

Academic Editor: Zhi-Yuan Sun

Copyright © 2018 Yafei Nie 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 CIP-MOC (Constrained Interpolation Profile/Method of Characteristics) is proposed to solve the tide wave equations with large time step size. The bottom topography and bottom friction, which are very important factor for the tidal wave model, are included to the equation of Riemann invariants as the source term. Numerical experiments demonstrate the good performance of the scheme. Compared to traditional semi-implicit (SI) finite difference scheme which is widely used in tidal wave simulation, CIP-MOC has better stability in simulating large gradient water surface change and has the ability to use much longer time step size under the premise of maintaining accuracy. Besides, numerical tests with reflective boundary conditions are carried out by CIP-MOC with large time step size and good results are obtained.

#### 1. Introduction

Tide wave is the water movement caused by the gravitational force of the celestial body. The control equations of tidal wave model are the original shallow water equations including source terms (e.g., bottom friction, bottom topography and horizontal eddy viscosity term) [1]. Therefore, getting the solution of shallow water equations with source terms is of great significance.

In the simulation of ocean tidal waves, Eulerian schemes are widely used, for example, Backhaus [2] and Casulli [3] used semi-implicit scheme (hereafter SI) for the solution of shallow water equations; Lv and Zhang [4] used the semi-implicit scheme to solve tide wave equations and their computational format was used to study bottom friction coefficients [5] and tidal open boundary conditions [6]; the finite volume method is also frequently used [7, 8]. However, since the phase speed of the gravity waves is so fast, time step is strictly limited by CFL condition when using Eulerian schemes [9].

There are growing interests in finding methods that enable using large time steps to solve shallow water equations. Erbes [9] proposed a semi-Lagrangian method of characteristics (MOC) with quadratic interpolation to solve the shallow water equations with a large CFL number, but there was dispersion error produced. Ogata and Yabe [10] applied the CIP method to shallow waters combined with the MOC and showed low dispersion error and low numerical damping even with large CFL number. Toda et al. [11] proposed a new scheme by adopting the Conservative Semi-Lagrangian (CSL) based on the CIP method which shows good conservation. However, applying this method in realistic simulation of ocean tidal wave will meet a difficulty treating source terms including bottom topography, bottom friction, and boundary conditions.

This paper is designated to apply the CIP-MOC [9] to the solution of tide wave equations which are based on shallow water equations and include source terms of bottom topography and friction. The ideal numerical experiment shows that the CIP-MOC method outperforms SI scheme in stability and dispersion errors when using large time step. Problem with reflection boundary condition is also discussed.

This paper is organized as follows. Section 2 presents the CIP-MOC method. In Section 3, numerical experiments are carried out and comparison is made between the CIP and SI. Conclusion is given in Section 4.

#### 2. Model and Method

##### 2.1. CIP Method

CIP method is a compact, robust, less diffusive, and high-order scheme in computational fluid dynamics [12]. Compared with other traditional interpolation methods, the internal information of grid is better described by CIP which uses spatial derivative in the description of function distribution in grid cells. We consider the following advection problem: the solution is [13]

If and are known at each grid point, the profile between two adjacent points can be approximated by cubic polynomialwhere . Consider four constraintsthen coefficients can be given as where , , and .

Thus, the profile of and at next time step can be obtained aswhere .

##### 2.2. One-Dimensional Tide Wave Equations

Let bottom friction term be the linear form, then the one-dimensional tide wave equations in a primitive form are written as [14]where is time, is Cartesian coordinate, is undisturbed water depth, is sea surface elevation above the undisturbed sea level, is the flow velocity, is the bottom friction coefficient, and is the gravitational acceleration.

To solve (9) with the method of characteristics and preserve the balance between the depth gradient and the bottom effect, the surface gradient method [15] is used here. Replace the variable with the water level , where represents the bottom topographic height (see Figure 1), we get the equivalent of (9) which can be written as