Solution of the linear shallow water equations by the fourth-order leapfrog scheme

1993 ◽  
Vol 98 (C6) ◽  
pp. 10205 ◽  
Author(s):  
Z. Kowalik
Keyword(s):  
Shallow Water ◽  
Fourth Order ◽  
Shallow Water Equations ◽  
Leapfrog Scheme

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

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.

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