Shock-Bubble Interaction Simulations Using a New Two-Phase Discontinuous Galerkin Method

Author(s):  
Leandro Gryngarten ◽  
Suresh Menon
2019 ◽  
Vol 11 (9) ◽  
pp. 168781401987490
Author(s):  
Muhammad Rehan Saleem ◽  
Ubaid Ahmed Nisar ◽  
Shamsul Qamar

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.


Geophysics ◽  
2021 ◽  
pp. 1-114
Author(s):  
Xijun He ◽  
Dinghui Yang ◽  
Yanjie Zhou ◽  
Yang Lei ◽  
xueyuan huang

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.


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