ON A STUDY OF STAGGERED LEAPFROG SCHEME FOR LINEAR SHALLOW WATER EQUATIONS

2020 ◽  
Vol 9 (11) ◽  
pp. 9787-9795
Author(s):  
L. Krismiyati Budiasih ◽  
L. Hari Wiryanto
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Wave Amplitude ◽  
Numerical Study ◽  
Shallow Water Equation ◽  
Tidal Basin ◽  
Constant Amplitude ◽  
Courant Number ◽  
Linear Friction ◽  
Leapfrog Scheme

In this paper, we present an analytical and numerical study of staggered leapfrog scheme for linear shallow water equation. It is shown that the scheme is stable when Courant number < 1, has second order accurate in both time and space, and there is no damping error in this scheme. We implement the scheme to simulate standing wave in a closed basin to show that the surface motions stay zero in a node and have constant amplitude at the antinode. For an external force given into the basin, it will induce a resonance, which cause the wave amplitude is getting bigger at the position of antinode. Moreover, we simulate a wave in a tidal basin, and show that the model has infinite spin up time. For a linear shallow water equation with linear friction, it is shown that the model has finite spin up time.

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.

scholarly journals On the standard Galerkin method with explicit RK4 time stepping for the shallow water equations

2019 ◽  
Vol 40 (4) ◽  
pp. 2415-2449
Author(s):  
D C Antonopoulos ◽  
V A Dougalis ◽  
G Kounadis
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Fourth Order ◽  
Shallow Water Equations ◽  
Computational Study ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Courant Number ◽  
Standard Galerkin Method ◽  
Initial Boundary Value

Abstract We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a quasiuniform mesh and in time by the classical four-stage, fourth order, explicit Runge–Kutta scheme. Assuming smoothness of solutions, a Courant number restriction and certain hypotheses on the finite element spaces, we prove $L^{2}$ error estimates that are of fourth-order accuracy in the temporal variable and of the usual, due to the nonuniform mesh, suboptimal order in space. We also make a computational study of the numerical spatial and temporal orders of convergence, and of the validity of a hypothesis made on the finite element spaces.

A modified leapfrog scheme for shallow water equations

2011 ◽  
Vol 52 ◽  
pp. 69-72 ◽  
Author(s):  
Wen-Yih Sun ◽  
Oliver M.T. Sun
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Leapfrog Scheme

scholarly journals Numerical Study of Spurious Inertial Modes in Shallow Water Models for a Variable Bathymetry

2019 ◽  
Vol 11 (6) ◽  
pp. 58
Author(s):  
Gossouhon SITIONON ◽  
Adama COULIBALY ◽  
Jérome Kablan ADOU
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Shallow Water Equations ◽  
Numerical Study ◽  
Ocean Modelling ◽  
Galerkin Approximations ◽  
Shallow Water Models ◽  
Water Models ◽  
Spurious Modes

In this study we perform a modal analysis of the linear inviscid shallow water equations using a non constant bathymetry, continuous and discontinuous Galerkin approximations. By extracting the discrete eigenvalues of the resulting algebraic linear system written on the form of a generalized eigenvalue / eigenvector problem we first show that the regular variation of the bathymetry does not prevent the presence of spurious inertial modes when centered finite element pairs are used. Secondly, we show that such spurious modes are not present in discontinuous Galerkin discretizations when all variables are approximated in the same descrete space. Such spurious inertial modes have been found very damageable for the quality of inertia-gravity and Rossby modes in ocean modelling.

Maximum power from a turbine farm in shallow water

2013 ◽  
Vol 714 ◽  
pp. 634-643 ◽  
Author(s):  
Chris Garrett ◽  
Patrick Cummins
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Momentum Equation ◽  
Basic Flow ◽  
Maximum Power ◽  
Two Dimensional ◽  
Dominant Term ◽  
Tidal Flows ◽  
Linear Friction

AbstractThe maximum power that can be obtained from a confined array of turbines in steady or tidal flows is considered using the two-dimensional shallow-water equations and representing the turbine farm by a uniform local increase in friction within a circle. Analytical results supported by dimensional reasoning and numerical solutions show that the maximum power depends on the dominant term in the momentum equation for flows perturbed on the scale of the farm. If friction dominates in the basic flow, the maximum power is a fraction (half for linear friction and 0.75 for quadratic friction) of the dissipation within the circle in the undisturbed state; if the advective terms dominate, the maximum power is a fraction of the undisturbed kinetic energy flux into the front of the turbine farm; if the acceleration dominates, the maximum power is similar to that for the linear frictional case, but with the friction coefficient replaced by twice the tidal frequency.

A NUMERICAL STUDY OF DAM-BREAK FLOW AND SEDIMENT TRANSPORT FROM A QUAKE LAKE

2011 ◽  
Vol 05 (05) ◽  
pp. 401-428 ◽  
Author(s):  
PENGZHI LIN ◽  
YINNA WU ◽  
JUNLI BAI ◽  
QUANHONG LIN
Keyword(s):  
Sediment Transport ◽  
Shallow Water ◽  
High Speed ◽  
Numerical Study ◽  
Shallow Water Equation ◽  
Equation Model ◽  
Dam Break ◽  
Bed Morphology ◽  
Dam Break Flow ◽  
Quake Lake

Dam-break flows are simulated numerically by a two-dimensional shallow-water-equation model that combines a hydrodynamic module and a sediment transport module. The model is verified by available analytical solutions and experimental data. It is demonstrated that the model is a reliable tool for the simulation of various transient shallow water flows and the associated sediment transport and bed morphology on complex topography. The validated model is then applied to investigate the potential dam-break flows from Tangjiashan Quake Lake resulting from Wenchuan Earthquake in 2008. The dam-break flow evolution is simulated by using the model in order to provide the flooding patterns (e.g., arrival time and flood height) downstream. Furthermore, the sediment transport and bed morphology simulation is performed locally to study the bed variation under the high-speed dam-break flow.

A Numerical Study for the Solution of Time Fractional Nonlinear Shallow Water Equation in Oceans

2013 ◽  
Vol 68 (8-9) ◽  
pp. 547-553 ◽  
Author(s):  
Sunil Kumar
Keyword(s):  
Analytical Solution ◽  
Shallow Water ◽  
Homotopy Perturbation Method ◽  
Numerical Study ◽  
Shallow Water Equation ◽  
Water System ◽  
General Characteristic ◽  
Convergent Series ◽  
Shallow Water Flows ◽  
Shallow Water System

In this paper, an analytical solution for the coupled one-dimensional time fractional nonlinear shallow water system is obtained by using the homotopy perturbation method (HPM). The shallow water equations are a system of partial differential equations governing fluid flow in the oceans (sometimes), coastal regions (usually), estuaries (almost always), rivers and channels (almost always). The general characteristic of shallow water flows is that the vertical dimension is much smaller than the typical horizontal scale. This method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. A very satisfactory approximate solution of the system with accuracy of the order 10-4 is obtained by truncating the HPM solution series at level six.

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