scholarly journals A Potential Enstrophy and Energy Conserving Scheme for the Shallow-Water Equations Extended to Generalized Curvilinear Coordinates

2017 ◽  
Vol 145 (3) ◽  
pp. 751-772 ◽  
Author(s):  
Michael D. Toy ◽  
Ramachandran D. Nair
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Empirical Test ◽  
Normal Mode Analysis ◽  
Second Order ◽  
Curvilinear Coordinates ◽  
Mode Analysis ◽  
Convergence Error ◽  
Starting Point ◽  
Potential Enstrophy

An energy and potential enstrophy conserving finite-difference scheme for the shallow-water equations is derived in generalized curvilinear coordinates. This is an extension of a scheme formulated by Arakawa and Lamb for orthogonal coordinate systems. The starting point for the present scheme is the shallow-water equations cast in generalized curvilinear coordinates, and tensor analysis is used to derive the invariant conservation properties. Preliminary tests on a flat plane with doubly periodic boundary conditions are presented. The scheme is shown to possess similar order-of-convergence error characteristics using a nonorthogonal coordinate compared to Cartesian coordinates for a nonlinear test of flow over an isolated mountain. A linear normal mode analysis shows that the discrete form of the Coriolis term provides stationary geostrophically balanced modes for the nonorthogonal coordinate and no unphysical computational modes are introduced. The scheme uses centered differences and averages, which are formally second-order accurate. An empirical test with a steady geostrophically balanced flow shows that the convergence rate of the truncation errors of the discrete operators is second order. The next step will be to adapt the scheme for use on the cubed sphere, which will involve modification at the lateral boundaries of the cube faces.

scholarly journals A Finite Volume Method for Modeling Shallow Flows with Wet-Dry Fronts on Adaptive Cartesian Grids

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
Sheng Bi ◽  
Jianzhong Zhou ◽  
Yi Liu ◽  
Lixiang Song
Keyword(s):  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Second Order ◽  
Topographic Features ◽  
Shallow Flows ◽  
Local Complex ◽  
Proposed Model ◽  
Volume Method

A second-order accurate, Godunov-type upwind finite volume method on dynamic refinement grids is developed in this paper for solving shallow-water equations. The advantage of this grid system is that no data structure is needed to store the neighbor information, since neighbors are directly specified by simple algebraic relationships. The key ingredient of the scheme is the use of the prebalanced shallow-water equations together with a simple but effective method to track the wet/dry fronts. In addition, a second-order spatial accuracy in space and time is achieved using a two-step unsplit MUSCL-Hancock method and a weighted surface-depth gradient method (WSDM) which considers the local Froude number is proposed for water depths reconstruction. The friction terms are solved by a semi-implicit scheme that can effectively prevent computational instability from small depths and does not invert the direction of velocity components. Several benchmark tests and a dam-break flooding simulation over real topography cases are used for model testing and validation. Results show that the proposed model is accurate and robust and has advantages when it is applied to simulate flow with local complex topographic features or flow conditions and thus has bright prospects of field-scale application.

scholarly journals Lattice Boltzmann method for 2D flows in curvilinear coordinates

2012 ◽  
Vol 14 (3) ◽  
pp. 772-783 ◽  
Author(s):  
Ljubomir Budinski
Keyword(s):  
Shallow Water ◽  
Lattice Boltzmann ◽  
Shallow Water Equations ◽  
Stokes Equations ◽  
Complex Geometry ◽  
Standard Form ◽  
Curvilinear Coordinates ◽  
Navier Stokes ◽  
Equilibrium Distribution Function ◽  
Rectangular Lattice

In order to improve efficiency and accuracy, while maintaining an ease of modeling flows with the lattice Boltzmann approach in domains having complex geometry, a method for modeling equations of 2D flow in curvilinear coordinates has been developed. Both the transformed shallow water equations and the transformed 2D Navier-Stokes equations in the horizontal plane were synchronized with the equilibrium distribution function and the force term in the rectangular lattice. Since the solution of these equations takes place in the classical rectangular lattice environment, boundary conditions are modeled in the standard form of already existing simple methods (bounce-back), not requiring any additional functions. Owing to this and to the fact that the proposed method ensures a more accurate fitting of equations, even to domains of interest having complex geometry, the accuracy of solution is significantly increased, while the simplicity of the standard lattice Boltzmann approach is maintained. For the shallow water equations transformed in curvilinear coordinates, the proposed procedure is verified in three different hydraulic problems, all characterized by complex geometry.

UNSPLITTING BGK-TYPE SCHEMES FOR THE SHALLOW WATER EQUATIONS

1999 ◽  
Vol 10 (04) ◽  
pp. 505-516 ◽  
Author(s):  
KUN XU
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Second Order ◽  
Time Dependent ◽  
Source Terms ◽  
Cell Interface

As a continuation of the research on BGK-type schemes, we present in this paper a second-order unsplitting method for the shallow water equations, where the source terms are included explicitly in the time-dependent flux functions across the cell interface.

Dam-break computations by a finite volume on dry regions

2021 ◽  
Author(s):  
Farid Boushaba ◽  
Salah Daoudi ◽  
Ahmed Yachouti ◽  
Youssef Regad
Keyword(s):  
Finite Elements ◽  
Shallow Water ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Shallow Water Equations ◽  
Equation Model ◽  
Second Order ◽  
Dam Break ◽  
Conservative Form ◽  
Volume Method

Abstract This paper presents numerical solvers, based on the finite volume method. This scheme solves dam break problems on the dry bottom in 2D configuration. The difficulty of the simulation of this type of problem lies in the propagation of shocks on the dry bottom. The equation model used is the shallow water equations written in conservative form. The scheme used is second order in space and time. The method is modified to treat dry bottoms. The validity of the method is demonstrated over the dam break example. A comparison with finite elements shows the weakness and robustness of each method.

Sign in / Sign up

Export Citation Format

Share Document