scholarly journals Enriched Galerkin method for the shallow-water equations

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Moritz Hauck ◽  
Vadym Aizinger ◽  
Florian Frank ◽  
Hennes Hajduk ◽  
Andreas Rupp
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Degrees Of Freedom ◽  
Piecewise Linear ◽  
Lower Number ◽  
Approximation Spaces ◽  
Continuous Galerkin ◽  
Dg Method ◽  
Implementation Aspects ◽  
Locally Conservative

AbstractThis work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similar to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic problems and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG.

An adaptive space-time shock capturing method with high order wavelet bases for the system of shallow water equations

2018 ◽  
Vol 28 (12) ◽  
pp. 2842-2861
Author(s):  
Hadi Minbashian ◽  
Hojatollah Adibi ◽  
Mehdi Dehghan
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Degrees Of Freedom ◽  
Adaptive Method ◽  
Space Time ◽  
High Order ◽  
Shock Capturing ◽  
Weak Formulation ◽  
Content Type ◽  
Wavelet Expansions

PurposeThis paper aims to propose an adaptive method for the numerical solution of the shallow water equations (SWEs). The authors provide an arbitrary high-order method using high-order spline wavelets. Furthermore, they use a non-linear shock capturing (SC) diffusion which removes the necessity of post-processing.Design/methodology/approachThe authors use a space-time weak formulation of SWEs which exploits continuous Galerkin (cG) in space and discontinuous Galerkin (dG) in time allowing time stepping, also known as cGdG. Such formulations along with SC term have recently been proved to ensure the stability of fully discrete schemes without scarifying the accuracy. However, the resulting scheme is expensive in terms of number of degrees of freedom (DoFs). By using natural adaptivity of wavelet expansions, the authors devise an adaptive algorithm to reduce the number of DoFs.FindingsThe proposed algorithm uses DoFs in a dynamic way to capture the shocks in all time steps while keeping the representation of approximate solution sparse. The performance of the proposed scheme is shown through some numerical examples.Originality/valueAn incorporation of wavelets for adaptivity in space-time weak formulations applied for SWEs is proposed.

scholarly journals Modified Predictor-Corrector WAF Method for the Shallow Water Equations with Source Terms

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Montri Maleewong
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Weighted Average ◽  
Test Cases ◽  
Half Time ◽  
Source Terms ◽  
Time Step ◽  
Predictor Corrector

A modified predictor-corrector scheme combining with the depth gradient method (DGM) and the weighted average flux (WAF) method has been presented to solve the one-dimensional shallow water equations with source terms. Approximate solutions in the predictor step are obtained by the DGM with piecewise-linear reconstructions in each cell volume. The source terms can then be calculated directly by these predicted values at the corresponding half-time step. In the corrector step, the TVD version of the WAF method is applied to calculate the numerical fluxes at the same half-time step for each cell face. The accuracy of numerical solutions is shown by applying the method to solve various test cases in both steady and unsteady problems with and without source terms. It shows that the numerical results are in good agreement with the existing analytical solutions as well as experimental data in some test cases.

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.

Sign in / Sign up

Export Citation Format

Share Document