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Author: Grafiati
Published: 4 May 2024

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1

Zhang, Rongpei, Xijun Yu, Jiang Zhu, Abimael F. D. Loula, and Xia Cui. "Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation." Communications in Computational Physics 14, no. 5 (November 2013): 1287–303. http://dx.doi.org/10.4208/cicp.190612.010313a.

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AbstractWeighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.
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2

Zhang, Rongpei, Xijun Yu, Mingjun Li, and Zhen Wang. "A semi-implicit integration factor discontinuous Galerkin method for the non-linear heat equation." Thermal Science 23, no. 3 Part A (2019): 1623–28. http://dx.doi.org/10.2298/tsci180921232z.

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In this paper, a new discontinuous Galerkin method is employed to study the non-linear heat conduction equation with temperature dependent thermal conductivity. We present practical implementation of the new discontinuous Galerkin scheme with weighted flux averages. The second-order implicit integration factor for time discretization method is applied to the semi discrete form. We obtain the L2 stability of the discontinuous Galerkin scheme. Numerical examples show that the error estimates are of second order when linear element approximations are applied. The method is applied to the non-linear heat conduction equations with source term.
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3

He, Xijun, Dinghui Yang, and Hao Wu. "A weighted Runge–Kutta discontinuous Galerkin method for wavefield modelling." Geophysical Journal International 200, no. 3 (January 24, 2015): 1389–410. http://dx.doi.org/10.1093/gji/ggu487.

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4

Liu, Yun-Long, Chi-Wang Shu, and A.-Man Zhang. "Weighted ghost fluid discontinuous Galerkin method for two-medium problems." Journal of Computational Physics 426 (February 2021): 109956. http://dx.doi.org/10.1016/j.jcp.2020.109956.

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5

Rustum, Ibrahim M., and ElHadi I. Elhadi. "Totally Volume Integral of Fluxes for Discontinuous Galerkin Method (TVI-DG) I-Unsteady Scalar One Dimensional Conservation Laws." Al-Mukhtar Journal of Sciences 32, no. 1 (June 30, 2017): 36–45. http://dx.doi.org/10.54172/mjsc.v32i1.124.

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The volume integral of Riemann flux in the discontinuous Galerkin (DG) method is introduced in this paper. The boundaries integrals of the fluxes (Riemann flux) are transformed into volume integral. The new family of DG method is accomplished by applying divergence theorem to the boundaries integrals of the flux. Therefore, the (DG) method is independent of the boundaries integrals of fluxes (Riemann flux) at the cell (element) boundaries as in classical (DG) methods. The modified streamline upwind Petrov-Galerkin method is used to capture the oscillation of unphysical flow for shocked flow problems. The numerical results of applying totally volume integral discontinuous Galerkin method (TVI-DG) are presented to unsteady scalar hyperbolic equations (linear convection equation, inviscid Burger's equation and Buckley-Leverett equation) for one dimensional case. The numerical finding of this scheme is very accurate as compared with other high order schemes as the weighted compact finite difference method WCOMP.
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6

Qiu, Chujun, Dinghui Yang, Xijun He, and Jingshuang Li. "A weighted Runge-Kutta discontinuous Galerkin method for reverse time migration." GEOPHYSICS 85, no. 6 (October 21, 2020): S343—S355. http://dx.doi.org/10.1190/geo2019-0193.1.

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Reverse time migration (RTM) is widely used in the industry because of its ability to handle complex geologic models including steeply dipping interfaces. The quality of images produced by RTM is significantly influenced by the performance of the numerical methods used to simulate the wavefields. Recently, a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method has been developed to solve the wave equation, which is stable, explicit, and efficient in parallelization and suppressing numerical dispersion. By incorporating two different weights for the time discretization, we have obtained a more stable method with a larger time sampling. We apply this numerical method to RTM to handle complex topography and improve imaging quality. By comparing it to the high-order Lax-Wendroff correction method, we determine that WRKDG is efficient in RTM. From the results of the Sigsbee2B data, we can find that our method is efficient in suppressing artifacts and can produce images of good quality when coarse meshes are used. The RTM results of the Canadian Foothills model also demonstrate its ability in handling complex geometry and rugged topography.
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7

Noels, L., and R. Radovitzky. "Alternative Approaches for the Derivation of Discontinuous Galerkin Methods for Nonlinear Mechanics." Journal of Applied Mechanics 74, no. 5 (July 17, 2006): 1031–36. http://dx.doi.org/10.1115/1.2712228.

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Discontinuous Galerkin methods are commonly derived by seeking a weak statement of the governing differential equations via a weighted-average approach allowing for discontinuous fields at the element interfaces of the discretization. In order to ensure consistency and stability of the formulation, this approach requires the definition of a numerical flux and a stabilization term. Discontinuous Galerkin methods may also be formulated from a linear combination of the governing and compatibility equations weighted by suitable operators. A third approach based on a variational statement of a generalized energy functional has been proposed recently for finite elasticity. This alternative approach naturally leads to an expression of the numerical flux and the stabilization terms in the context of large deformation mechanics problems. This paper compares these three approaches and establishes the conditions under which identical formulations are obtained.
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8

Zhu, Jun, and Jianxian Qiu. "Runge-Kutta Discontinuous Galerkin Method Using Weno-Type Limiters: Three-Dimensional Unstructured Meshes." Communications in Computational Physics 11, no. 3 (March 2012): 985–1005. http://dx.doi.org/10.4208/cicp.300810.240511a.

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AbstractThis paper further considers weighted essentially non-oscillatory (WENO) and Hermite weighted essentially non-oscillatory (HWENO) finite volume methods as limiters for Runge-Kutta discontinuous Galerkin (RKDG) methods to solve problems involving nonlinear hyperbolic conservation laws. The application discussed here is the solution of 3-D problems on unstructured meshes. Our numerical tests again demonstrate this is a robust and high order limiting procedure, which simultaneously achieves high order accuracy and sharp non-oscillatory shock transitions.
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9

Bassonon, Yibour Corentin, and Arouna Ouedraogo. "Discontinuous Galerkin method for linear parabolic equations with L^1-data." Gulf Journal of Mathematics 16, no. 2 (April 12, 2024): 122–34. http://dx.doi.org/10.56947/gjom.v16i2.1874.

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In this work, we examine the discontinuous Galerkin method for parabolic linear problem with data in L1(Ω × (0, T)). On one hand, using a Euler time advancing scheme that goes backwards, we can discretize a time interval. Furthermore, the discretization of space is based on Symmetric Weighted Interior Penalty (SWIPG) method. We use the technique of construction of the renormalized solution to obtain existence of the discrete solution. Then, our research demonstrates that the discrete solution converges in L1(Q) to the unique renormalized solution of the problem, where Q= Ω × (0, T). In the case where the coefficients are smooth, we offer an estimate of the error in L1(Q), when the side on the right is assigned to Marcinkiewicz space Ls, ∞(Q) where 1 < s < 2.
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10

Zhang, Fan, Tiegang Liu, and Moubin Liu. "A third-order weighted variational reconstructed discontinuous Galerkin method for solving incompressible flows." Applied Mathematical Modelling 91 (March 2021): 1037–60. http://dx.doi.org/10.1016/j.apm.2020.10.011.

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11

Shu, Chi-Wang. "On high-order accurate weighted essentially non-oscillatory and discontinuous Galerkin schemes for compressible turbulence simulations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 371, no. 1982 (January 13, 2013): 20120172. http://dx.doi.org/10.1098/rsta.2012.0172.

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In this article, we give a brief overview on high-order accurate shock capturing schemes with the aim of applications in compressible turbulence simulations. The emphasis is on the basic methodology and recent algorithm developments for two classes of high-order methods: the weighted essentially non-oscillatory and discontinuous Galerkin methods.
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12

Abdulle, Assyr, and Giacomo Rosilho de Souza. "A local discontinuous Galerkin gradient discretization method for linear and quasilinear elliptic equations." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 4 (July 2019): 1269–303. http://dx.doi.org/10.1051/m2an/2019022.

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A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic problems in high gradient regions. Using the gradient discretization framework we prove convergence of the scheme for linear and quasilinear equations under minimal regularity assumptions. The error due to artificial boundary conditions is also analyzed, shown to be of higher order and shown to depend only locally on the regularity of the solution. Numerical experiments illustrate our theoretical findings and the local method’s accuracy is compared against the non local approach.
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13

Maday, Yvon, and Carlo Marcati. "Regularity and hp discontinuous Galerkin finite element approximation of linear elliptic eigenvalue problems with singular potentials." Mathematical Models and Methods in Applied Sciences 29, no. 08 (July 2019): 1585–617. http://dx.doi.org/10.1142/s0218202519500295.

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We study the regularity in weighted Sobolev spaces of Schrödinger-type eigenvalue problems, and we analyze their approximation via a discontinuous Galerkin (dG) [Formula: see text] finite element method. In particular, we show that, for a class of singular potentials, the eigenfunctions of the operator belong to analytic-type non-homogeneous weighted Sobolev spaces. Using this result, we prove that an isotropically graded [Formula: see text] dG method is spectrally accurate, and that the numerical approximation converges with exponential rate to the exact solution. Numerical tests in two and three dimensions confirm the theoretical results and provide an insight into the behavior of the method for varying discretization parameters.
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14

Zhong, Xinghui, and Chi-Wang Shu. "A simple weighted essentially nonoscillatory limiter for Runge–Kutta discontinuous Galerkin methods." Journal of Computational Physics 232, no. 1 (January 2013): 397–415. http://dx.doi.org/10.1016/j.jcp.2012.08.028.

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15

Chen, Tsu-Fen, Hyesuk Lee, and Chia-Chen Liu. "Numerical approximation of the Oldroyd-B model by the weighted least-squares/discontinuous Galerkin method." Numerical Methods for Partial Differential Equations 29, no. 2 (March 30, 2012): 531–48. http://dx.doi.org/10.1002/num.21719.

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16

Hoteit, Hussein, and Abbas Firoozabadi. "Compositional Modeling by the Combined Discontinuous Galerkin and Mixed Methods." SPE Journal 11, no. 01 (March 1, 2006): 19–34. http://dx.doi.org/10.2118/90276-pa.

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Summary In this work, we present a numerical procedure that combines the mixed finite-element (MFE) and the discontinuous Galerkin (DG) methods. This numerical scheme is used to solve the highly nonlinear coupled equations that describe the flow processes in homogeneous and heterogeneous media with mass transfer between the phases. The MFE method is used to approximate the phase velocity based on the pressure (more precisely average pressure) at the interface between the nodes. This approach conserves the mass locally at the element level and guarantees the continuity of the total flux across the interfaces. The DG method is used to solve the mass-balance equations, which are generally convection-dominated. The DG method associated with suitable slope limiters can capture sharp gradients in the solution without creating spurious oscillations. We present several numerical examples in homogeneous and heterogeneous media that demonstrate the superiority of our method to the finite-difference (FD) approach. Our proposed MFE-DG method becomes orders of magnitude faster than the FD method for a desired accuracy in 2D. Introduction There has been gradual progress in the development of algorithms for the compositional simulation of hydrocarbon reservoirs in the last 15 years. Before that, there were several major advances in the numerical solution of the combined flow equations and the thermodynamic equilibrium with the equations of state. Despite the advances of the last 25 to 30 years and the enormous progress in the speed of computers in the same period, we cannot yet perform field-scale compositional modeling satisfactorily in heterogeneous reservoirs. The main problem is the continued use of the FD discretization scheme and its inherent limitations. Most of the current compositional simulators use the upstream weighted FD method to approximate the flow equations. Because of the fact that the flow processes are usually convection-dominated, FD methods may produce significant numerical diffusion (Coats 1980). The excessive numerical diffusion requires unrealistic gridding, especially with heterogeneities. Recently, the DG methods have been successfully implemented to approximate various physical problems, notably hyperbolic systems of conservative laws. One property of these methods is that they conserve mass at the element level in a finite-element framework. Consequently, they enhance the flexibility of finite elements in describing flow in complicated geometries. Furthermore, the choice of the spatial approximation without the continuity across inter-element boundaries allows a simple treatment of combined finite-element cells with different geometries as well as different degrees of approximating polynomials. These methods associated with suitable slope limiters can capture discontinuities or sharp gradients in the solution. The DG method was first implemented for nonlinear scalar conservative laws by Chavent and Salzano (1982). However, these authors noted that a very restrictive timestep should be used to keep stability of the scheme.
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17

Hossain, Muhammad Shakhawat, Chunguang Xiong, and Huafei Sun. "A priori and a posteriori error analysis of the first order hyperbolic equation by using DG method." PLOS ONE 18, no. 3 (March 30, 2023): e0277126. http://dx.doi.org/10.1371/journal.pone.0277126.

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In this research article, a discontinuous Galerkin method with a weighted parameter θ and a penalty parameter γ is proposed for solving the first order hyperbolic equation. The key aim of this method is to design an error estimation for both a priori and a posteriori error analysis on general finite element meshes. It is also exposed to the reliability and effectiveness of both parameters in the order of convergence of the solutions. For a posteriori error estimation, residual adaptive mesh- refining algorithm is employed. A series of numerical experiments are illustrated that demonstrate the efficiency of the method.
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18

Yang, Dinghui, Xijun He, Xiao Ma, Yanjie Zhou, and Jingshuang Li. "An optimal nearly analytic discrete-weighted Runge-Kutta discontinuous Galerkin hybrid method for acoustic wavefield modeling." GEOPHYSICS 81, no. 5 (September 2016): T251—T263. http://dx.doi.org/10.1190/geo2015-0686.1.

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The newly developed optimal nearly analytic discrete (ONAD) and the weighted Runge-Kutta discontinuous Galerkin (WRKDG) methods can effectively suppress the numerical dispersion caused by discretizing wave equations, but it is difficult for ONAD to implement on flexible meshes, whereas the WRKDG has high computational cost for wavefield simulations. We have developed a new hybrid algorithm by combining the ONAD method with the WRKDG method. In this hybrid algorithm, the computational domain was split into several subdomains, in which the subdomain for the ONAD method used regular Cartesian grids, whereas the subdomain for the WRKDG method used triangular grids. The hybrid method was at least third-order spatially accurate. We have applied the proposed method to simulate the scalar wavefields for different models, including a homogeneous model, a rough topography model, a fracture model, and a cave model. The numerical results found that the hybrid method can deal with complicated geometrical structures, effectively suppress numerical dispersion, and provide accurate seismic wavefields. Numerical examples proved that our hybrid method can significantly reduce the CPU time and save storage requirement for the tested models. This implies that the hybrid method is especially suitable for the simulation of waves propagating in complex media.
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19

Sharma, Dipty, and Paramjeet Singh. "Discontinuous Galerkin approximation for excitatory-inhibitory networks with delay and refractory periods." International Journal of Modern Physics C 31, no. 03 (January 30, 2020): 2050041. http://dx.doi.org/10.1142/s0129183120500412.

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In this study, we consider the network of noisy leaky integrate-and-fire (NNLIF) model, which governs by a second-order nonlinear time-dependent partial differential equation (PDE). This equation uses the probability density approach to describe the behavior of neurons with refractory states and the transmission delays. A numerical approximation based on the discontinuous Galerkin (DG) method is used for the spatial discretization with the analysis of stability. The strong stability-preserving explicit Runge–Kutta (SSPERK) method is performed for the temporal discretization. Finally, some test examples and numerical simulations are given to examine the behavior of the solution. The execution of the constructed scheme is measured by the quantitative comparison with the existing finite difference technique, namely weighted essentially nonoscillatory (WENO) scheme.
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20

Zhang, Yifan, and Ramachandran D. Nair. "A Nonoscillatory Discontinuous Galerkin Transport Scheme on the Cubed Sphere." Monthly Weather Review 140, no. 9 (September 1, 2012): 3106–26. http://dx.doi.org/10.1175/mwr-d-11-00287.1.

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Abstract The discontinuous Galerkin (DG) method is high order, conservative, and offers excellent parallel efficiency. However, when there are discontinuities in the solution, the DG transport scheme generates spurious oscillations that reduce the quality of the numerical solution. For applications such as the atmospheric tracer transport modeling, a nonoscillatory, positivity-preserving solution is a basic requirement. To suppress the oscillations in the DG solution, a limiter based on the Hermite-Weighted Essentially Nonoscillatory (H-WENO) method has been implemented for a third-order DG transport scheme. However, the H-WENO limiter can still produce wiggles with small amplitudes in the solutions, but this issue has been addressed by combining the limiter with a bound-preserving (BP) filter. The BP filter is local and easy to implement and can be used for making the solution strictly positivity preserving. The DG scheme combined with the limiter and filter preserves the accuracy of the numerical solution in the smooth regions while effectively eliminating overshoots and undershoots. The resulting nonoscillatory DG scheme is third-order accurate (P2-DG) and based on the modal discretization. The 2D Cartesian scheme is further extended to the cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates. The accuracy and effectiveness of the limiter and filter are demonstrated with several benchmark tests on both the Cartesian and spherical geometries.
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21

Zhu, Jun, Xinghui Zhong, Chi-Wang Shu, and Jianxian Qiu. "Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter." Communications in Computational Physics 19, no. 4 (April 2016): 944–69. http://dx.doi.org/10.4208/cicp.070215.200715a.

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AbstractIn this paper, we propose a new type of weighted essentially non-oscillatory (WENO) limiter, which belongs to the class of Hermite WENO (HWENO) limiters, for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving hyperbolic conservation laws. This new HWENO limiter is a modification of the simple WENO limiter proposed recently by Zhong and Shu [29]. Both limiters use information of the DG solutions only from the target cell and its immediate neighboring cells, thus maintaining the original compactness of the DG scheme. The goal of both limiters is to obtain high order accuracy and non-oscillatory properties simultaneously. The main novelty of the new HWENO limiter in this paper is to reconstruct the polynomial on the target cell in a least square fashion [8] while the simple WENO limiter [29] is to use the entire polynomial of the original DG solutions in the neighboring cells with an addition of a constant for conservation. The modification in this paper improves the robustness in the computation of problems with strong shocks or contact discontinuities, without changing the compact stencil of the DG scheme. Numerical results for both one and two dimensional equations including Euler equations of compressible gas dynamics are provided to illustrate the viability of this modified limiter.
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22

Pongsanguansin, Thida, Montri Maleewong, and Khamron Mekchay. "Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows." Modelling and Simulation in Engineering 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/591282.

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A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.
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23

Führer, Thomas, Norbert Heuer, and Jhuma Sen Gupta. "A Time-Stepping DPG Scheme for the Heat Equation." Computational Methods in Applied Mathematics 17, no. 2 (April 1, 2017): 237–52. http://dx.doi.org/10.1515/cmam-2016-0037.

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AbstractWe introduce and analyze a discontinuous Petrov–Galerkin method with optimal test functions for the heat equation. The scheme is based on the backward Euler time stepping and uses an ultra-weak variational formulation at each time step. We prove the stability of the method for the field variables (the original unknown and its gradient weighted by the square root of the time step) and derive a Céa-type error estimate. For low-order approximation spaces this implies certain convergence orders when time steps are not too small in comparison with mesh sizes. Some numerical experiments are reported to support our theoretical results.
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24

Botti, Michele, Daniele A. Di Pietro, and Pierre Sochala. "A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity." Computational Methods in Applied Mathematics 20, no. 2 (April 1, 2020): 227–49. http://dx.doi.org/10.1515/cmam-2018-0142.

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AbstractIn this work, we construct and analyze a nonconforming high-order discretization method for the quasi-static single-phase nonlinear poroelasticity problem describing Darcean flow in a deformable porous medium saturated by a slightly compressible fluid. The nonlinear elasticity operator is discretized using a Hybrid High-Order method, while the Darcy operator relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, supports arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle both nonzero and vanishing specific storage coefficients.
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25

Li, Wanai, Qian Wang, and Yu-Xin Ren. "A p-weighted limiter for the discontinuous Galerkin method on one-dimensional and two-dimensional triangular grids." Journal of Computational Physics 407 (April 2020): 109246. http://dx.doi.org/10.1016/j.jcp.2020.109246.

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26

Ern, A., A. F. Stephansen, and P. Zunino. "A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity." IMA Journal of Numerical Analysis 29, no. 2 (April 2, 2008): 235–56. http://dx.doi.org/10.1093/imanum/drm050.

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27

Jamei, Mehdi, and H. Ghafouri. "An efficient discontinuous Galerkin method for two-phase flow modeling by conservative velocity projection." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 1 (January 4, 2016): 63–84. http://dx.doi.org/10.1108/hff-08-2014-0247.

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Purpose – The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection. Design/methodology/approach – The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations. Findings – The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements. Research limitations/implications – The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible. Practical implications – The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media. Originality/value – The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.
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28

Zhu, Jun, Xinghui Zhong, Chi-Wang Shu, and Jianxian Qiu. "Runge-Kutta Discontinuous Galerkin Method with a Simple and Compact Hermite WENO Limiter on Unstructured Meshes." Communications in Computational Physics 21, no. 3 (February 7, 2017): 623–49. http://dx.doi.org/10.4208/cicp.221015.160816a.

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AbstractIn this paper we generalize a new type of compact Hermite weighted essentially non-oscillatory (HWENO) limiter for the Runge-Kutta discontinuous Galerkin (RKDG) method, which was recently developed in [38] for structured meshes, to two dimensional unstructured meshes. The main idea of this HWENO limiter is to reconstruct the new polynomial by the usage of the entire polynomials of the DG solution from the target cell and its neighboring cells in a least squares fashion [11] while maintaining the conservative property, then use the classical WENO methodology to form a convex combination of these reconstructed polynomials based on the smoothness indicators and associated nonlinear weights. The main advantage of this new HWENO limiter is the robustness for very strong shocks and simplicity in implementation especially for the unstructured meshes considered in this paper, since only information from the target cell and its immediate neighbors is needed. Numerical results for both scalar and system equations are provided to test and verify the good performance of this new limiter.
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29

Mei, Yanjie, Sulei Wang, Zhijie Xu, Chuanjing Song, and Yao Cheng. "Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems." Symmetry 13, no. 12 (December 1, 2021): 2291. http://dx.doi.org/10.3390/sym13122291.

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We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.
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30

Li, Wanai, and Yang Liu. "The p-Weighted Limiter for the Discontinuous Galerkin Method in Solving Compressible Flows on Tetrahedral Grids." International Journal of Computational Fluid Dynamics 35, no. 7 (August 9, 2021): 510–33. http://dx.doi.org/10.1080/10618562.2021.2003789.

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31

Qiu, Jing-Mei, and Chi-Wang Shu. "Convergence of High Order Finite Volume Weighted Essentially Nonoscillatory Scheme and Discontinuous Galerkin Method for Nonconvex Conservation Laws." SIAM Journal on Scientific Computing 31, no. 1 (January 2008): 584–607. http://dx.doi.org/10.1137/070687487.

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32

Zunino, Paolo. "Discontinuous Galerkin Methods Based on Weighted Interior Penalties for Second Order PDEs with Non-smooth Coefficients." Journal of Scientific Computing 38, no. 1 (July 17, 2008): 99–126. http://dx.doi.org/10.1007/s10915-008-9219-3.

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Liu, XiaoJing, Xueshang Feng, Man Zhang, and Jingmin Zhao. "Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme." Astrophysical Journal Supplement Series 265, no. 1 (March 1, 2023): 19. http://dx.doi.org/10.3847/1538-4365/acb14f.

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Abstract The present study aims to apply an implicit high-order reconstructed discontinuous Galerkin (DG) scheme (rDG(P 1 P 2)) to simulate the steady-state solar corona. In this scheme, a piecewise quadratic polynomial solution, P 2, is obtained from the underlying piecewise linear DG solution, P 1, by least-squares reconstruction with a weighted essentially nonoscillatory limiter. The reconstructed quadratic polynomial solution is then used for the computation of the fluxes and source terms. In addition, an implicit time integration method with large time steps is considered in this work. The resulting large linear algebraic system of equations from the implicit discretization is solved by the cellwise relaxation implicit scheme which can make full use of the compactness of the DG scheme. The code of the implicit high-order rDG scheme is developed in Fortran language with message passing interface parallelization in Cartesian coordinates. To validate this code, we first test a problem with an exact solution, which confirms the expected third-order accuracy. Then we simulate the solar corona for Carrington rotations 2167, 2183, and 2210, and compare the modeled results with observations. We find that the numerical results basically reproduce the large-scale observed structures of the solar corona, such as coronal holes, helmet streamers, pseudostreamers, and high- and low-speed streams, which demonstrates the capability of the developed scheme.
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34

Pei, Weicheng, Yuyan Jiang, and Shu Li. "High-Order CFD Solvers on Three-Dimensional Unstructured Meshes: Parallel Implementation of RKDG Method with WENO Limiter and Momentum Sources." Aerospace 9, no. 7 (July 11, 2022): 372. http://dx.doi.org/10.3390/aerospace9070372.

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In aerospace engineering, high-order computational fluid dynamics (CFD) solvers suitable for three-dimensional unstructured meshes are less developed than expected. The Runge–Kutta discontinuous Galerkin (RKDG) finite element method with compact weighted essentially non-oscillatory (WENO) limiters is one of the candidates, which might give high-order solutions on unstructured meshes. In this article, we provide an efficient parallel implementation of this method for simulating inviscid compressible flows. The implemented solvers are tested on lower-dimensional model problems and real three-dimensional engineering problems. Results of lower-dimensional problems validate the correctness and accuracy of these solvers. The capability of capturing complex flow structures even on coarse meshes is shown in the results of three-dimensional applications. For solving problems containing rotary wings, an unsteady momentum source model is incorporated into the solvers. Such a finite element/momentum source hybrid method eliminates the reliance on advanced mesh techniques, which might provide an efficient tool for studying rotorcraft aerodynamics.
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35

Blaise, Sébastien, and Amik St-Cyr. "A Dynamic hp-Adaptive Discontinuous Galerkin Method for Shallow-Water Flows on the Sphere with Application to a Global Tsunami Simulation." Monthly Weather Review 140, no. 3 (March 1, 2012): 978–96. http://dx.doi.org/10.1175/mwr-d-11-00038.1.

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Abstract A discontinuous Galerkin model solving the shallow-water equations on the sphere is presented. It captures the dynamically varying key aspects of the flows by having the advantageous ability to locally modify the mesh as well as the order of interpolation within each element. The computational load is efficiently distributed among processors in parallel using a weighted recursive coordinate bisection strategy. A simple error estimator, based on the discontinuity of the variables at the interfaces between elements, is used to select the elements to be refined or coarsened. The flows are expressed in three-dimensional Cartesian coordinates, but tangentially constrained to the sphere by adding a Lagrange multiplier to the system of equations. The model is validated on classic atmospheric test cases and on the simulation of the February 2010 Chilean tsunami propagation. The proposed multiscale strategy is able to reduce the computational time by an order of magnitude on the tsunami simulation, clearly demonstrating its potential toward multiresolution three-dimensional oceanic and atmospheric applications.
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36

Qiu, Jianxian, and Chi-Wang Shu. "A Comparison of Troubled-Cell Indicators for Runge--Kutta Discontinuous Galerkin Methods Using Weighted Essentially Nonoscillatory Limiters." SIAM Journal on Scientific Computing 27, no. 3 (January 2005): 995–1013. http://dx.doi.org/10.1137/04061372x.

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37

Jamei, Mehdi, and H. Ghafouri. "A novel discontinuous Galerkin model for two-phase flow in porous media using an improved IMPES method." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 1 (January 4, 2016): 284–306. http://dx.doi.org/10.1108/hff-01-2015-0008.

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Purpose – The purpose of this paper is to present an efficient improved version of Implicit Pressure-Explicit Saturation (IMPES) method for the solution of incompressible two-phase flow model based on the discontinuous Galerkin (DG) numerical scheme. Design/methodology/approach – The governing equations, based on the wetting-phase pressure-saturation formulation, are discretized using various primal DG schemes. The authors use H(div) velocity reconstruction in Raviart-Thomas space (RT_0 and RT_1), the weighted average formulation, and the scaled penalties to improve the spatial discretization. It uses a new improved IMPES approach, by using the second-order explicit Total Variation Diminishing Runge-Kutta (TVD-RK) as temporal discretization of the saturation equation. The main purpose of this time stepping technique is to speed up computation without losing accuracy, thus to increase the efficiency of the method. Findings – Utilizing pressure internal interpolation technique in the improved IMPES scheme can reduce CPU time. Combining the TVD property with a strong multi-dimensional slope limiter namely, modified Chavent-Jaffre leads to a non-oscillatory scheme even in coarse grids and highly heterogeneous porous media. Research limitations/implications – The presented locally conservative scheme can be applied only in 2D incompressible two-phase flow modeling in non-deformable porous media. In addition, the capillary pressure discontinuity between two adjacent rock types assumed to be negligible. Practical implications – The proposed numerical scheme can be efficiently used to model the incompressible two-phase flow in secondary recovery of petroleum reservoirs and tracing immiscible contamination in aquifers. Originality/value – The paper describes a novel version of the DG two-phase flow which illustrates the effects of improvements in special discretization. Also the new improved IMPES approach used reduces the computation time. The non-oscillatory scheme is an efficient algorithm as it maintains accuracy and saves computation time.
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Rafiei, Behnam, Hamed Masoumi, Mohammad Saeid Aghighi, and Amine Ammar. "Effects of complex boundary conditions on natural convection of a viscoplastic fluid." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 8 (August 5, 2019): 2792–808. http://dx.doi.org/10.1108/hff-09-2018-0507.

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Purpose The purpose of this paper is to analyze the effects of complex boundary conditions on natural convection of a yield stress fluid in a square enclosure heated from below (uniformly and non-uniformly) and symmetrically cooled from the sides. Design/methodology/approach The governing equations are solved numerically subject to continuous and discontinuous Dirichlet boundary conditions by Galerkin’s weighted residuals scheme of finite element method and using a non-uniform unstructured triangular grid. Findings Results show that the overall heat transfer from the heated wall decreases in the case of non-uniform heating for both Newtonian and yield stress fluids. It is found that the effect of yield stress on heat transfer is almost similar in both uniform and non-uniform heating cases. The yield stress has a stabilizing effect, reducing the convection intensity in both cases. Above a certain value of yield number Y, heat transfer is only due to conduction. It is found that a transition of different modes of stability may occur as Rayleigh number changes; this fact gives rise to a discontinuity in the variation of critical yield number. Originality/value Besides the new numerical method based on the finite element and using a non-uniform unstructured grid for analyzing natural convection of viscoplastic materials with complex boundary conditions, the originality of the present work concerns the treatment of the yield stress fluids under the influence of complex boundary conditions.
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Gopalakrishnan, J., and G. Kanschat. "A multilevel discontinuous Galerkin method." Numerische Mathematik 95, no. 3 (September 1, 2003): 527–50. http://dx.doi.org/10.1007/s002110200392.

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Gopalakrishnan, J., and G. Kanschat. "A multilevel discontinuous Galerkin method." Numerische Mathematik 95, no. 3 (September 1, 2003): 551. http://dx.doi.org/10.1007/s00211-003-0504-7.

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41

Xu, Liyang, Xinhai Xu, Xiaoguang Ren, Yunrui Guo, Yongquan Feng, and Xuejun Yang. "Stability evaluation of high-order splitting method for incompressible flow based on discontinuous velocity and continuous pressure." Advances in Mechanical Engineering 11, no. 10 (October 2019): 168781401985558. http://dx.doi.org/10.1177/1687814019855586.

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In this work, we deal with high-order solver for incompressible flow based on velocity correction scheme with discontinuous Galerkin discretized velocity and standard continuous approximated pressure. Recently, small time step instabilities have been reported for pure discontinuous Galerkin method, in which both velocity and pressure are discretized by discontinuous Galerkin. It is interesting to examine these instabilities in the context of mixed discontinuous Galerkin–continuous Galerkin method. By means of numerical investigation, we find that the discontinuous Galerkin–continuous Galerkin method shows great stability at the same configuration. The consistent velocity divergence discretization scheme helps to achieve more accurate results at small time step size. Since the equal order discontinuous Galerkin–continuous Galerkin method does not satisfy inf-sup stability requirement, the instability for high Reynolds number flow is investigated. We numerically demonstrate that fine mesh resolution and high polynomial order are required to obtain a robust system. With these conclusions, discontinuous Galerkin–continuous Galerkin method is able to achieve high-order spatial convergence rate and accurately simulate high Reynolds flow. The solver is tested through a series of classical benchmark problems, and efficiency improvement is proved against pure discontinuous Galerkin scheme.
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42

Hu, Qingjie, Yinnian He, Tingting Li, and Jing Wen. "A Mixed Discontinuous Galerkin Method for the Helmholtz Equation." Mathematical Problems in Engineering 2020 (May 4, 2020): 1–9. http://dx.doi.org/10.1155/2020/9582583.

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In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.
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43

Walkington, Noel J. "Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions." SIAM Journal on Numerical Analysis 42, no. 5 (January 2005): 1801–17. http://dx.doi.org/10.1137/s0036142902412233.

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44

Richter, Gerard R. "The discontinuous Galerkin method with diffusion." Mathematics of Computation 58, no. 198 (May 1, 1992): 631. http://dx.doi.org/10.1090/s0025-5718-1992-1122076-2.

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45

Lai, Wencong, and Abdul A. Khan. "Time stepping in discontinuous Galerkin method." Journal of Hydrodynamics 25, no. 3 (June 2013): 321–29. http://dx.doi.org/10.1016/s1001-6058(11)60370-4.

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46

Freund, Jouni. "The space-continuous–discontinuous Galerkin method." Computer Methods in Applied Mechanics and Engineering 190, no. 26-27 (March 2001): 3461–73. http://dx.doi.org/10.1016/s0045-7825(00)00279-6.

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47

Liu, Xiaodong, Nathaniel R. Morgan, and Donald E. Burton. "A Lagrangian discontinuous Galerkin hydrodynamic method." Computers & Fluids 163 (February 2018): 68–85. http://dx.doi.org/10.1016/j.compfluid.2017.12.007.

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48

Allen, Myron B., and George F. Pinder. "The Convergence of Upstream Collocation in the Buckley-Leverett Problem(includes associated papers 14810 and 14970 )." Society of Petroleum Engineers Journal 25, no. 03 (June 1, 1985): 363–70. http://dx.doi.org/10.2118/10978-pa.

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Abstract Upstream collocation is a fast and accurate scheme for simulating multiphase flows in oil reservoirs. In contrast to standard orthogonal collocation, upstream collocation yields numerical solutions to the Buckley-Leverett problem that converge to correct solutions physically. The failure of standard orthogonal collocation is not surprising, since the Buckley-Leverett problem as commonly stated is posed incompletely. The equal-area rule of Buckley and Leverett and the Welge tangent construction both specify additional constraints needed to close the problem properly. An error analysis of upstream collocation shows that this method forces convergence through an artificial dissipative term analogous to the "vanishing viscosity" used in shock fitting. This constraint is mathematically equivalent to the more familiar constructions and should prove beneficial in stimulating EOR schemes based on frontal displacement. Introduction The Buckley-Leverett saturation equation is of fundamental importance in the mechanics of oil recovery, yet solving the equation poses difficulties when saturation shocks are present. Analytic or graphic methods must negotiate triple-valued saturations, while naively applied numerical solutions may yield incorrect solutions. Orthogonal collocation is a noteworthy example: it conserves mass but, like centered difference schemes, misplaces shocks. All these problems reflect the fact that the usual statements of Cauchy problems for the Buckley-Leverett saturation equation are incomplete. To guarantee uniqueness of discontinuous solutions requires, in addition to initial data, the specification of a shock condition that is mathematically equivalent to several physically reasonable constraints. The Buckley-Leverett equal-area rule and Welge's tangent construction both implement this extra condition. A recently developed numerical method called upstream collocation overcomes the convergence failures of orthogonal collocation, generating solutions with steep gradients at the correct shock location. The intent of this paper is to demonstrate that upstream collocation enforces the correct shock condition through an error term that mimics dissipation but vanishes on refinement of the spatial grid. This error term parallels the lowest-order error terms in upstream-weighted finite differences and achieves the same effects as the artificial capillary pressures used in several earlier finite-element formulations. The analysis leading to the form of the error rests on a correspondence between collocation and Galerkin schemes and follows a line of reasoning originally developed for the linear, parabolic convection-diffusion equation. The gist of the argument is that upstream collocation corresponds to an erroneous approximation of the integrals arising in Galerkin's method. Calculation of the quadrature error leads to an expression for the artificial dissipation induced by upstream collocation. To clarify why such an error term is appropriate, we precede the error analysis with a brief review of the physical setting, solutions, and mathematics of the Buckley-Leverett problem. Physical Setting The Buckley-Leverett saturation equation describes the simultaneous flow of two immiscible, incompressible fluids through a homogeneous porous medium. The equation is important in oil production because fluid injection and displacement are common to essentially all EOR schemes. In simple applications the displaced fluid is oil, and the displacing fluid may be either water or gas. The equation arises from a material balance on water and the two-phase extension of Darcy's law, as Buckley and Leverett describe in their original paper. For a one-dimensional reservoir with uniform rock properties, combining this set of governing equations yields ............................(1) where S is water saturation, q is the effective total flow rate [L/T], and f(s) signifies the fraction of flowing fluid that is water. We assume a consistent set of units. In a horizontal, vertically uniform reservoir, the fractional flow function off is related to saturation-dependent rock and fluid properties as follows: ............................(2) where (S) and (S) are the oil and water mobilities, respectively, and(S) is the capillary pressure. SPEJ P. 363^
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Zhang, Xiao, Xiaoping Xie, and Shiquan Zhang. "An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems." Computational Methods in Applied Mathematics 19, no. 4 (October 1, 2019): 849–61. http://dx.doi.org/10.1515/cmam-2018-0007.

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AbstractThe embedded discontinuous Galerkin (EDG) method by Cockburn, Gopalakrishnan and Lazarov [B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 47 2009, 2, 1319–1365] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second-order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees {k+1}, {k+1}, k ({k\geq 0}) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.
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Liu, Donghuan, and Yinghua Liu. "Applications of Discontinuous Galerkin Finite Element Method in Thermomechanical Coupling Problems with Imperfect Thermal Contact." Mathematical Problems in Engineering 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/861417.

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Thermomechanical coupling problems with imperfect thermal contact are analyzed in the present paper with discontinuous Galerkin finite element method. The imperfect thermal contact condition is characterized by thermal contact resistance. The whole thermomechanical coupling problem is solved alternatively with the thermal subproblem and mechanical subproblem. Thermal contact resistance is introduced directly with the interface numerical flux of the present discontinuous Galerkin finite element method without using interface element as traditional continuous Galerkin finite element method does. Numerical results show the accuracy and feasibility of the present discontinuous Galerkin finite element method in solving thermomechanical coupling problems with imperfect thermal contact.
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