Well-Balanced Numerical Schemes for Shallow Water Equations with Horizontal Temperature Gradient

2018 ◽  
Vol 43 (1) ◽  
pp. 783-807 ◽  
Author(s):  
Mai Duc Thanh ◽  
Nguyen Xuan Thanh
Keyword(s):  
Temperature Gradient ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Horizontal Temperature Gradient ◽  
Numerical Schemes

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

2015 ◽  
Vol 2015 ◽  
pp. 1-10
Author(s):  
Xinhua Lu ◽  
Bingjiang Dong ◽  
Bing Mao ◽  
Xiaofeng Zhang
Keyword(s):  
High Resolution ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Small Time ◽  
Numerical Schemes ◽  
Time Step ◽  
Numerical Diffusion ◽  
First Order ◽  
Small Time Step ◽  
The One

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.

High Order Two Dimensional Numerical Schemes for the Coupling of Transport Equations and Shallow Water Equations

2008 ◽  
pp. 241-248
Author(s):  
M. J. Castro ◽  
E. D. Fernández Nieto ◽  
A. M. Ferreiro Ferreiro ◽  
J. A. García Rodríguez ◽  
C. Parés
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Transport Equations ◽  
High Order ◽  
Two Dimensional ◽  
Numerical Schemes

scholarly journals Deficiency of finite difference methods for capturing shock waves and wave propagation over uneven bottom seabed

2018 ◽  
Vol 7 (3.28) ◽  
pp. 97
Author(s):  
Mohammad Fadhli Ahmad ◽  
Mohd Sofiyan Suliman ◽  
. .
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Source Term ◽  
Finite Difference Methods ◽  
Dam Break ◽  
Numerical Schemes ◽  
Inaccurate Information ◽  
Uneven Bottom

The implementation of finite difference method is used to solve shallow water equations under the extreme conditions. The cases such as dam break and wave propagation over uneven bottom seabed are selected to test the ordinary schemes of Lax-Friederichs and Lax-Wendroff numerical schemes. The test cases include the source term for wave propagation and exclude the source term for dam break. The main aim of this paper is to revisit the application of Lax-Friederichs and Lax-Wendroff numerical schemes at simulating dam break and wave propagation over uneven bottom seabed. For the case of the dam break, the two steps of Lax-Friederichs scheme produce non-oscillation numerical results, however, suffering from some of dissipation. Moreover, the two steps of Lax-Wendroff scheme suffers a very bad oscillation. It seems that these numerical schemes cannot solve the problem at discontinuities which leads to oscillation and dissipation. For wave propagation case, those numerical schemes produce inaccurate information of free surface and velocity due to the uneven seabed profile. Therefore, finite difference is unable to model shallow water equations under uneven bottom seabed with high accuracy compared to the analytical solution.  

scholarly journals Flood wave propagation in steep mountain rivers

2012 ◽  
Vol 15 (1) ◽  
pp. 120-137 ◽  
Author(s):  
Gabriella Petaccia ◽  
Luigi Natale ◽  
Fabrizio Savi ◽  
Mirjana Velickovic ◽  
Yves Zech ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Source Term ◽  
Term Treatment ◽  
Flood Wave ◽  
Numerical Schemes ◽  
Steady State Condition ◽  
One Dimensional ◽  
The One

Most of the recent developments concerning efficient numerical schemes to solve the shallow-water equations in view of real world flood modelling purposes concern the two-dimensional form of the equations or the one-dimensional form written for rectangular, unit-width channels. Extension of these efficient schemes to the one-dimensional cross-sectional averaged shallow-water equations is not straightforward, especially when complex natural topographies are considered. This paper presents different formulations of numerical schemes based on the HLL (Harten–Lax–van Leer) solver, and the adaptation of the topographical source term treatment when cross-sections of arbitrary shape are considered. Coupled and uncoupled formulations of the equations are considered, in combination with centred and lateralised source term treatment. These schemes are compared to a numerical solver of Lax Friedrichs type based on a staggered grid. The proposed schemes are first tested against two theoretical benchmark tests and then applied to the Brembo River, an Italian alpine river, firstly simulating a steady-state condition and secondly reproducing the 2002 flood wave propagation.

scholarly journals SEMI-IMPLICIT NUMERICAL SCHEMA IN SHALLOW WATER EQUATION

2017 ◽  
Vol 17 (2) ◽  
pp. 102
Author(s):  
Safwandi Safwandi ◽  
Syamsul Rizal ◽  
Tarmizi Tarmizi
Keyword(s):  
New York ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Shallow Water Equation ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Spectral Difference ◽  
Spectral Difference Method

Abstract. A two-dimensional shallow water equation integrated on depth water based on finite differential methods. Numerical solutions with different methods consist of explicit, implicit and semi-implicit schemes. Different methods of shallow water equations expressed in numerical schemes. For bottom-friction is described in semi-implicitly. This scheme will be more flexible for initial values and boundary conditions when compared to the explicit schemes.  Keywords: 2D numerical models, shallow water equations, explicit and semi-implicit schema.Reference Hassan, H. S., Ramadan, K. T., Hanna, S. N. 2010. Numerical Solution of the Rotating Shallow Water Flows with Topography Using the Fractional Steps Method, Scie.Res,App.Math. (1):104-117. Omer, S, Kursat, K. 2011. High-Order Accurate Spectral Difference Method For Shallow Water Equations. IJRRAS6. Vol. 6. No. 1. Kampf, J. 2009. Ocean Modelling for Beginners. Springer Heidelberg Dordrecht. London, New York. Wang, Z. L., Geng, Y. F. 2013. Two-Dimensional Shallow Water Equations with Porosity and Their Numerical scheme on Unstructured Grids. J. Water Science and Engineering. Vol. 6, No. 1, 91-105. Saiduzzaman, Sobuj. 2013. Comparison of Numerical Schemes for Shallow Water Equation. Global J. of Sci. Fron. Res. Math. and Dec. Sci. Vol. 13 (4). Sari, C. I., Surbakti, H., Fauziyah., Pola Sebaran Salinatas dengan Model Numerik Dua Dimensi di Muara Sungai Musi. Maspari J. Vol. 5 (2): 104-110. Bunya, B., Westerink, J. J. dan Shinobu, Y. 2004. Discontinuous Boundary Implementation for the Shallow Water Equations. Int. J. Numer. Meth. Fluids 2005 (47): 1451–1468. 

scholarly journals Instability in Leapfrog and Forward–Backward Schemes

2010 ◽  
Vol 138 (5) ◽  
pp. 1497-1501 ◽  
Author(s):  
Wen-Yih Sun
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Schemes ◽  
Courant Number ◽  
Diffusion Term ◽  
Repeated Eigenvalues ◽  
Leapfrog Scheme

Abstract This paper shows that in the linearized shallow-water equations, the numerical schemes can become weakly unstable for the 2Δx wave in the C grid when the Courant number is 1 in the forward–backward scheme and 0.5 in the leapfrog scheme because of the repeated eigenvalues in the matrices. The instability can be amplified and spread to other waves and smaller Courant number if the diffusion term is included. However, Shuman smoothing can control the instability.

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