Simulations of shallow water equations by finite difference WENO schemes with multilevel time discretization

2020 ◽  
Author(s):  
Dwisunu Prayitno ◽  
Sumardi Sumardi
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Time Discretization ◽  
Weno Schemes ◽  
Finite Difference Weno Schemes

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

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Changna Lu ◽  
Luoyan Xie ◽  
Hongwei Yang
Keyword(s):  
Finite Difference ◽  
Shallow Water ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Bottom Topography ◽  
Time Discretization ◽  
Weno Scheme ◽  
Weighted Essentially Nonoscillatory ◽  
Accuracy Order ◽  
Essentially Nonoscillatory

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.

RECENT DEVELOPMENTS IN THE NUMERICAL SIMULATION OF SHALLOW WATER EQUATIONS II: TEMPORAL DISCRETIZATION

1994 ◽  
Vol 04 (04) ◽  
pp. 533-556 ◽  
Author(s):  
V. AGOSHKOV ◽  
E. OVCHINNIKOV ◽  
A. QUARTERONI ◽  
F. SALERI
Keyword(s):  
Numerical Simulation ◽  
Finite Element ◽  
Finite Difference ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Results ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Recent Developments ◽  
Numerical Approaches

This paper deals with time-advancing schemes for shallow water equations. We review some of the existing numerical approaches, propose new schemes and investigate their stability. We present numerical results obtained using the time-advancing schemes proposed, with finite element and finite difference approximation in space variables.

A new type of finite difference WENO schemes for Hamilton–Jacobi equations

2019 ◽  
Vol 30 (02n03) ◽  
pp. 1950020 ◽  
Author(s):  
Xiaohan Cheng ◽  
Jianhu Feng ◽  
Supei Zheng ◽  
Xueli Song
Keyword(s):  
Finite Difference ◽  
Weno Schemes ◽  
Numerical Examples ◽  
Weighted Essentially Nonoscillatory ◽  
New Type ◽  
Spurious Oscillations ◽  
Symmetry Requirement ◽  
Finite Difference Weno Schemes ◽  
Jacobi Equations ◽  
Fifth Order

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

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