scholarly journals Discontinuous Galerkin method for solving incompressible two-phase shallow granular flow model

2019 ◽  
Vol 11 (9) ◽  
pp. 168781401987490
Author(s):  
Muhammad Rehan Saleem ◽  
Ubaid Ahmed Nisar ◽  
Shamsul Qamar
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Scheme ◽  
Flow Model ◽  
Numerical Study ◽  
Momentum Exchange ◽  
Two Phase ◽  
Runge Kutta ◽  
Model Equations

This article deals with the numerical study of two-phase shallow flow model describing the mixture of fluid and solid granular particles. The model under investigation consists of coupled mass and momentum equations for solid granular material and fluid particles through non-conservative momentum exchange terms. The non-conservativity of model equations poses major challenges for any numerical scheme, such as well balancing, positivity preservation, accurate approximation of non-conservative terms, and achievement of steady-state conditions. Thus, in order to approximate the present model an accurate, well-balanced, robust, and efficient numerical scheme is required. For this purpose, in this article, Runge–Kutta discontinuous Galerkin method is applied successfully for the first time to solve the model equations. Several test problems are also carried out to check the performance and accuracy of our proposed numerical method. To compare the results, the same model is solved by staggered central Nessyahu–Tadmor scheme. A good comparison is found between two schemes, but the results obtained by Runge–Kutta discontinuous Galerkin scheme are found superior over the central Nessyahu–Tadmor scheme.

Runge-Kutta discontinuous Galerkin method for solving wave equations in 2D isotropic and anisotropic poroelastic media at low frequencies

Geophysics ◽  
2021 ◽  
pp. 1-114
Author(s):  
Xijun He ◽  
Dinghui Yang ◽  
Yanjie Zhou ◽  
Yang Lei ◽  
xueyuan huang
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Wave Equations ◽  
Order System ◽  
Elastic Model ◽  
Two Phase ◽  
Low Frequencies ◽  
Poroelastic Media ◽  
Runge Kutta

A Runge-Kutta discontinuous Galerkin (RKDG) method for solving wave equations in isotropic and anisotropic poroelastic media at low frequencies is introduced. First, the 2D Biot’s two-phase equations are transformed into a first-order system with dissipation. Then, the system is discretized by using the discontinuous Galerkin method (DGM) with a third-order Runge-Kutta time discretization. The numerical stability conditions for solving porous equations are also investigated. We test several examples to validate the proposed method in isotropic and anisotropic poroelastic media. Comparisons of seismic responses with the finite-difference method (FDM) on fine grids show the correctness of this method. Moreover, the numerical results indicate that the RKDG method can provide clear fast P, slow P and S waves for anisotropic poroelastic media on coarse meshes. Also, a two-layer porous model, a poroelastic-elastic model with horizontal interface and an isotropic-anisotropic poroelastic model with sinusoidal interface demonstrate that the proposed method can deal with complex wave propagation. Therefore, the simulation results show that the RKDG method is an accurate and stable method for solving Biot’s equations.

Runge–Kutta discontinuous Galerkin method for Baer–Nunziato model with ‘simple WENO’ limiting of conservative variables

2021 ◽  
Vol 36 (2) ◽  
pp. 57-74
Author(s):  
Mikhail Alekseev ◽  
Evgeny Savenkov
Keyword(s):  
Mathematical Model ◽  
Galerkin Method ◽  
Numerical Simulations ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Numerical Algorithm ◽  
Hyperbolic Model ◽  
Two Phase Flows ◽  
Two Phase ◽  
Runge Kutta

Abstract The work is devoted to the application of Runge–Kutta discontinuous Galerkin (RKDG) method for solving Baer–Nunziato hyperbolic model for nonequilibrium two-phase flows. The approach is based on the application of the simple WENO limiter directly to the conservative variables. Mathematical model and the corresponding numerical algorithm are described. The results of numerical simulations for 1D and 2D tests are presented and discussed.

An efficient discontinuous Galerkin method on triangular meshes for a pedestrian flow model

2008 ◽  
Vol 76 (3) ◽  
pp. 337-350 ◽  
Author(s):  
Yinhua Xia ◽  
S. C. Wong ◽  
Mengping Zhang ◽  
Chi-Wang Shu ◽  
William H. K. Lam
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Flow Model ◽  
Pedestrian Flow ◽  
Triangular Meshes

Runge-Kutta Discontinuous Galerkin Method with Front Tracking Method for Solving the Compressible Two-Medium Flow on Unstructured Meshes

2016 ◽  
Vol 9 (1) ◽  
pp. 73-91 ◽  
Author(s):  
Haitian Lu ◽  
Jun Zhu ◽  
Chunwu Wang ◽  
Ning Zhao
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Front Tracking ◽  
Unstructured Meshes ◽  
Interfacial Region ◽  
Tracking Method ◽  
Front Tracking Method ◽  
Medium Flow ◽  
Runge Kutta

AbstractIn this paper, we extend using the Runge-Kutta discontinuous Galerkin method together with the front tracking method to simulate the compressible two-medium flow on unstructured meshes. A Riemann problem is constructed in the normal direction in the material interfacial region, with the goal of obtaining a compact, robust and efficient procedure to track the explicit sharp interface precisely. Extensive numerical tests including the gas-gas and gas-liquid flows are provided to show the proposed methodologies possess the capability of enhancing the resolutions nearby the discontinuities inside of the single medium flow and the interfacial vicinities of the two-medium flow in many occasions.

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