scholarly journals Third‐ and fourth‐order well‐balanced schemes for the shallow water equations based on the CWENO reconstruction

2018 ◽  
Vol 89 (8) ◽  
pp. 304-325 ◽  
Author(s):  
Manuel J. Castro ◽  
Matteo Semplice
Keyword(s):  
Shallow Water ◽  
Fourth Order ◽  
Shallow Water Equations ◽  
Cweno Reconstruction

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.

MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF COASTAL ACOUSTIC TOMOGRAPHY

2017 ◽  
Author(s):  
Xiao-Hua Zhu ◽  
Xiao-Hua Zhu ◽  
Ze-Nan Zhu ◽  
Ze-Nan Zhu ◽  
Xinyu Guo ◽  
...  
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Mean Amplitude ◽  
Momentum Equation ◽  
Tidal Currents ◽  
Acoustic Tomography ◽  
Mean Values ◽  
Residual Currents ◽  
Water Depths ◽  
Deep Area

A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to 13, 2009. The horizontal distributions of tidal current in the tomography domain are calculated by the inverse analysis in which the travel time differences for sound traveling reciprocally are used as data. Spatial mean amplitude ratios M2 : M4 : M6 are 1.00 : 0.15 : 0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, velocity amplitudes of M4 measured by CAT agree well with those of M4 predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area where water depths are larger than 60 m is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. Dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents.

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