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Academic literature on the topic 'Weighted Discontinuous Galerkin method'

Author: Grafiati

Published: 4 May 2024

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Journal articles on the topic "Weighted Discontinuous Galerkin method"

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

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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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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 (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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Qiu, Chujun, Dinghui Yang, Xijun He, and Jingshuang Li. "A weighted Runge-Kutta discontinuous Galerkin method for reverse time migration." GEOPHYSICS 85, no. 6 (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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Noels, L., and R. Radovitzky. "Alternative Approaches for the Derivation of Discontinuous Galerkin Methods for Nonlinear Mechanics." Journal of Applied Mechanics 74, no. 5 (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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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 (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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Bassonon, Yibour Corentin, and Arouna Ouedraogo. "Discontinuous Galerkin method for linear parabolic equations with L^1-data." Gulf Journal of Mathematics 16, no. 2 (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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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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Dissertations / Theses on the topic "Weighted Discontinuous Galerkin method"

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Johansson, August. "Duality-based adaptive finite element methods with application to time-dependent problems." Doctoral thesis, Umeå : Institutionen för matematik och matematisk statistik, Umeå universitet, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-33872.

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Marcati, Carlo. "Discontinuous hp finite element methods for elliptic eigenvalue problems with singular potentials : with applications to quantum chemistry." Thesis, Sorbonne université, 2018. http://www.theses.fr/2018SORUS349.

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Dans cette thèse, on étudie des problèmes aux valeurs propres elliptiques avec des potentiels singuliers, motivés par plusieurs modèles en physique et en chimie quantique, et on propose une méthode des éléments finis de type hp discontinus (dG) adaptée pour l’approximation des modes propres. Dans ces modèles, arrivent naturellement des potentiels singuliers (associés à l’interaction entre noyaux et électrons). Notre analyse commence par une étude de la régularité elliptique dans des espaces de Sobolev à poids. On montre comment un opérateur elliptique avec potentiel singulier est un isomorphisme entre espaces de Sobolev à poids non homogènes et que l’on peut développer des bornes de type analytique à poids sur les solutions des problèmes aux valeurs propres associés aux opérateurs. La méthode hp/dG graduée qu’on utilise converge ainsi de façon exponentielle. On poursuit en considérant une classe de problèmes non linéaires représentatifs des applications. On montre que, sous certaines conditions, la méthode hp/dG graduée converge et que, si la non linéarité est de type polynomiale, on obtient les mêmes estimations de type analytique que dans le cas linéaire. De plus, on étudie la convergence de la valeur propre pour voir sous quelles conditions la vitesse de convergence est améliorée par rapport à celle des vecteurs propres. Pour tous les cas considérés, on effectue des tests numériques, qui ont pour objectif à la fois de valider les résultats théoriques, mais aussi d’évaluer le rôle des sources d’erreur non considérées dans l’analyse et d’aider dans la conception de méthode hp/dG graduée pour des problèmes plus complexes<br>In this thesis, we study elliptic eigenvalue problems with singular potentials, motivated by several models in physics and quantum chemistry, and we propose a discontinuous Galerkin hp finite element method for their solution. In these models, singular potentials occur naturally (associated with the interaction between nuclei and electrons). Our analysis starts from elliptic regularity in non homogeneous weighted Sobolev spaces. We show that elliptic operators with singular potential are isomorphisms in those spaces and that we can derive weighted analytic type estimates on the solutions to the linear eigenvalue problems. The isotropically graded hp method provides therefore approximations that converge with exponential rate to the solution of those eigenproblems. We then consider a wide class of nonlinear eigenvalue problems, and prove the convergence of numerical solutions obtained with the symmetric interior penalty discontinuous Galerkin method. Furthermore, when the non linearity is polynomial, we show that we can obtain the same analytic type estimates as in the linear case, thus the numerical approximation converges exponentially. We also analyze under what conditions the eigenvalue converges at an increased rate compared to the eigenfunctions. For both the linear and nonlinear case, we perform numerical tests whose objective is both to validate the theoretical results, but also evaluate the role of sources of errors not considered previously in the analysis, and to help in the design of hp/dG graded methods for more complex problems

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Gürkan, Ceren. "Extended hybridizable discontinuous Galerkin method." Doctoral thesis, Universitat Politècnica de Catalunya, 2018. http://hdl.handle.net/10803/664035.

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This thesis proposes a new numerical technique: the eXtended Hybridizable Discontinuous Galerkin (X-HDG) Method, to efficiently solve problems including moving boundaries and interfaces. It aims to outperform available methods and improve the results by inheriting favored properties of Discontinuous Galerkin (HDG) together with an explicit interface definition. X-HDG combines the Hybridizable HDG method with an eXtended Finite Element (X-FEM) philosophy, with a level set description of the interface, to form an hp convergent, high order unfitted numerical method. HDG outperforms other Discontinuous Galerkin (DG) methods for problems involving self-adjoint operators, due to its hybridization and superconvergence properties. The hybridization process drastically reduces the number of degrees of freedom in the discrete problem, similarly to static condensation in the context of high-order Continuous Galerkin (CG). On other hand, HDG is based on a mixed formulation that, differently to CG or other DG methods, is stable even when all variables (primal unknowns and derivatives) are approximated with polynomials of the same degree k. As a result, convergence of order k+1 in the L2 norm is proved not only for the primal unknown, but also for its derivatives. Therefore, a simple element-by-element postprocess of the derivatives leads to a superconvergent approximation of the primal variables, with convergence of order k+2 in the L2 norm. X-HDG inherits these favored properties of HDG in front of CG and DG methods; moreover, thanks to the level set description of interfaces, costly remeshing is avoided when dealing with moving interfaces. This work demonstrates that X-HDG keeps the optimal and superconvergence of HDG with no need of mesh fitting to the interface. In Chapters 2 and 3, the X-HDG method is derived and implemented to solve the steady-state Laplace equation on a domain where the interface separates a single material from the void and where the interface separates two different materials. The accuracy and the convergence of X-HDG is tested over examples with manufactured solutions and it is shown that X-HDG outperforms the previous proposals by demonstrating high order optimum and super convergence, together with reduced system size thanks to its hybrid nature, without mesh fitting. In Chapters 4 and 5, the X-HDG method is derived and implemented to solve Stokes interface problem for void and bimaterial interfaces. With X-HDG, high order convergence is demonstrated over unfitted meshes for incompressible flow problems. X-HDG for moving interfaces is studied in Chapter 6. A transient Laplace problem is considered, where the time dependent term is discretized using the backward Euler method. A collapsing circle example together with two-phase Stefan problem are analyzed in numerical examples section. It is demonstrated that X-HDG offers high-order optimal convergence for time-dependent problems. Moreover, with Stefan problem, using a polynomial degree k, a more accurate approximation of interface position is demonstrated against X-FEM, thanks to k+1 convergent gradient approximation of X-HDG. Yet again, results obtained by previous proposals are improved.<br>Esta tesis propone una nueva técnica numérica: eXtended Hybridizable Discontinuous Galerkin (X-HDG), para resolver eficazmente problemas incluyendo fronteras en movimiento e interfaces. Su objetivo es superar las limitaciones de los métodos disponibles y mejorar los resultados, heredando propiedades del método Hybridizable Discontinuous Galerkin method (HDG), junto con una definición de interfaz explícita. X-HDG combina el método HDG con la filosofía de eXtended Finite Element method (X-FEM), con una descripción level-set de la interfaz, para obtener un método numérico hp convergente de orden superior sin ajuste de la malla a la interfaz o frontera. HDG supera a otros métodos de DG para los problemas implícitos con operadores autoadjuntos, debido a sus propiedades de hibridación y superconvergencia. El proceso de hibridación reduce drásticamente el número de grados de libertad en el problema discreto, similar a la condensación estática en el contexto de Continuous Galerkin (CG) de alto orden. Por otro lado, HDG se basa en una formulación mixta que, a diferencia de CG u otros métodos DG, es estable incluso cuando todas las variables (incógnitas primitivas y derivadas) se aproximan con polinomios del mismo grado k. Como resultado, la convergencia de orden k + 1 en la norma L2 se demuestra no sólo para la incógnita primal sino también para sus derivadas. Por lo tanto, un simple post-proceso elemento-a-elemento de las derivadas conduce a una aproximación superconvergente de las variables primales, con convergencia de orden k+2 en la norma L2. X-HDG hereda estas propiedades. Por otro lado, gracias a la descripción level-set de la interfaz, se evita caro remallado tratando las interfaces móviles. Este trabajo demuestra que X-HDG mantiene la convergencia óptima y la superconvergencia de HDG sin la necesidad de ajustar la malla a la interfaz. En los capítulos 2 y 3, se deduce e implementa el método X-HDG para resolver la ecuación de Laplace estacionaria en un dominio donde la interfaz separa un solo material del vacío y donde la interfaz separa dos materiales diferentes. La precisión y convergencia de X-HDG se prueba con ejemplos de soluciones fabricadas y se demuestra que X-HDG supera las propuestas anteriores mostrando convergencia óptima y superconvergencia de alto orden, junto con una reducción del tamaño del sistema gracias a su naturaleza híbrida, pero sin ajuste de la malla. En los capítulos 4 y 5, el método X-HDG se desarrolla e implementa para resolver el problema de interfaz de Stokes para interfaces vacías y bimateriales. Con X-HDG, de nuevo se muestra una convergencia de alto orden en mallas no adaptadas, para problemas de flujo incompresible. X-HDG para interfaces móviles se discute en el Capítulo 6. Se considera un problema térmico transitorio, donde el término dependiente del tiempo es discretizado usando el método de backward Euler. Un ejemplo de una interfaz circulas que se reduce, junto con el problema de Stefan de dos fases, se discute en la sección de ejemplos numéricos. Se demuestra que X-HDG ofrece un alto grado de convergencia óptima para problemas dependientes del tiempo. Además, con el problema de Stefan, usando un grado polinomial k, se demuestra una aproximación más exacta de la posición de la interfaz contra X-FEM, gracias a la aproximación del gradiente convergente k + 1 de X-HDG. Una vez más, se mejoran los resultados obtenidos por las propuestas anteriores

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Kaufmann, Willem. "Extended Hydrodynamics Using the Discontinuous-Galerkin Hancock Method." Thesis, Université d'Ottawa / University of Ottawa, 2021. http://hdl.handle.net/10393/42672.

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Moment methods derived from the kinetic theory of gases can be used for the prediction of continuum and non-equilibrium flows and offer numerical advantages over other methods, such as the Navier-Stokes model. Models developed in this fashion are described by first-order hyperbolic partial differential equations (PDEs) with stiff local relaxation source terms. The application of discontinuous-Galerkin (DG) methods for the solution of such models has many benefits. Of particular interest is the third-order accurate, coupled space-time discontinuous-Galerkin Hancock (DGH) method. This scheme is accurate, as well as highly efficient on large-scale distributed-memory computers. The current study outlines a general implementation of the DGH method used for the parallel solution of moment methods in one, two, and three dimensions on modern distributed clusters. An algorithm for adaptive mesh refinement (AMR) was developed alongside the implementation of the scheme, and is used to achieve even higher accuracy and efficiency. Many different first-order hyperbolic and hyperbolic-relaxation PDEs are solved to demonstrate the robustness of the scheme. First, a linear convection-relaxation equation is solved to verify the order of accuracy of the scheme in three dimensions. Next, some classical compressible Euler problems are solved in one, two, and three dimensions to demonstrate the scheme's ability to capture discontinuities and strong shocks, as well as the efficacy of the implemented AMR. A special case, Ringleb's flow, is also solved in two-dimensions to verify the order of accuracy of the scheme for non-linear PDEs on curved meshes. Following this, the shallow water equations are solved in two dimensions. Afterwards, the ten-moment (Gaussian) closure is applied to two-dimensional Stokes flow past a cylinder, showing the abilities of both the closure and scheme to accurately compute classical viscous solutions. Finally, the one-dimensional fourteen-moment closure is solved.

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Toprakseven, Suayip. "Error Analysis of Extended Discontinuous Galerkin (XdG) Method." University of Cincinnati / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1418733307.

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Elfverson, Daniel. "Discontinuous Galerkin Multiscale Methods for Elliptic Problems." Thesis, Uppsala universitet, Institutionen för informationsteknologi, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-138960.

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In this paper a continuous Galerkin multiscale method (CGMM) and a discontinuous Galerkin multiscale method (DGMM) are proposed, both based on the variational multiscale method for solving partial differential equations numerically. The solution is decoupled into a coarse and a fine scale contribution, where the fine-scale contribution is computed on patches with localized right hand side. Numerical experiments are presented where exponential decay of the error is observed when increasing the size of the patches for both CGMM and DGMM. DGMM gives much better accuracy when the same size of the patches are used.

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Galbraith, Marshall C. "A Discontinuous Galerkin Chimera Overset Solver." University of Cincinnati / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1384427339.

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Lui, Ho Man. "Runge-Kutta Discontinuous Galerkin method for the Boltzmann equation." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/39215.

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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2006.<br>Includes bibliographical references (p. 85-87).<br>In this thesis we investigate the ability of the Runge-Kutta Discontinuous Galerkin (RKDG) method to provide accurate and efficient solutions of the Boltzmann equation. Solutions of the Boltzmann equation are desirable in connection to small scale science and technology because when characteristic flow length scales become of the order of, or smaller than, the molecular mean free path, the Navier-Stokes description fails. The prevalent Boltzmann solution method is a stochastic particle simulation scheme known as Direct Simulation Monte Carlo (DSMC). Unfortunately, DSMC is not very effective in low speed flows (typical of small scale devices of interest) because of the high statistical uncertainty associated with the statistical sampling of macroscopic quantities employed by this method. This work complements the recent development of an efficient low noise method for calculating the collision integral of the Boltzmann equation, by providing a high-order discretization method for the advection operator balancing the collision integral in the Boltzmann equation. One of the most attractive features of the RKDG method is its ability to combine high-order accuracy, both in physical space and time, with the ability to capture discontinuous solutions.<br>(cont.) The validity of this claim is thoroughly investigated in this thesis. It is shown that, for a model collisionless Boltzmann equation, high-order accuracy can be achieved for continuous solutions; whereas for discontinuous solutions, the RKDG method, with or without the application of a slope limiter such as a viscosity limiter, displays high-order accuracy away from the vicinity of the discontinuity. Given these results, we developed a RKDG solution method for the Boltzmann equation by formulating the collision integral as a source term in the advection equation. Solutions of the Boltzmann equation, in the form of mean velocity and shear stress, are obtained for a number of characteristic flow length scales and compared to DSMC solutions. With a small number of elements and a low order of approximation in physical space, the RKDG method achieves similar results to the DSMC method. When the characteristic flow length scale is small compared to the mean free path (i.e. when the effect of collisions is small), oscillations are present in the mean velocity and shear stress profiles when a coarse velocity space discretization is used. With a finer velocity space discretization, the oscillations are reduced, but the method becomes approximately five times more computationally expensive.<br>(cont.) We show that these oscillations (due to the presence of propagating discontinuities in the distribution function) can be removed using a viscosity limiter at significantly smaller computational cost.<br>by Ho Man Lui.<br>S.M.

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Bala, Chandran Ram. "Development of discontinuous Galerkin method for nonlocal linear elasticity." Thesis, Massachusetts Institute of Technology, 2007. http://hdl.handle.net/1721.1/41730.

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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.<br>Includes bibliographical references (p. 75-81).<br>A number of constitutive theories have arisen describing materials which, by nature, exhibit a non-local response. The formulation of boundary value problems, in this case, leads to a system of equations involving higher-order derivatives which, in turn, results in requirements of continuity of the solution of higher order. Discontinuous Galerkin methods are particularly attractive toward this end, as they provide a means to naturally enforce higher interelement continuity in a weak manner without the need of modifying the finite element interpolation. In this work, a discontinuous Galerkin formulation for boundary value problems in small strain, non-local linear elasticity is proposed. The underlying theory corresponds to the phenomenological strain-gradient theory developed by Fleck and Hutchinson within the Toupin-Mindlin framework. The single-field displacement method obtained enables the discretization of the boundary value problem with a conventional continuous interpolation inside each finite element, whereas the higher-order interelement continuity is enforced in a weak manner. The proposed method is shown to be consistent and stable both theoretically and with suitable numerical examples.<br>by Ram Bala Chandran.<br>S.M.

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Ekström, Sven-Erik. "A vertex-centered discontinuous Galerkin method for flow problems." Licentiate thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-284321.

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The understanding of flow problems, and finding their solution, has been important for most of human history, from the design of aqueducts to boats and airplanes. The use of physical miniature models and wind tunnels were, and still are, useful tools for design, but with the development of computers, an increasingly large part of the design process is assisted by computational fluid dynamics (CFD). Many industrial CFD codes have their origins in the 1980s and 1990s, when the low order finite volume method (FVM) was prevalent. Discontinuous Galerkin methods (DGM) have, since the turn of the century, been seen as the successor of these methods, since it is potentially of arbitrarily high order. In its lowest order form DGM is equivalent to FVM. However, many existing codes are not compatible with standard DGM and would need a complete rewrite to obtain the advantages of the higher order. This thesis shows how to extend existing vertex-centered and edge-based FVM codes to higher order, using a special kind of DGM discretization, which is different from the standard cell-centered type. Two model problems are examined to show the necessary data structures that need to be constructed, the order of accuracy for the method, and the use of an hp-adaptation scheme to resolve a developing shock. Then the method is further developed to solve the steady Euler equations, within the existing industrial Edge code, using acceleration techniques such as local time stepping and multigrid. With the ever increasing need for more efficient and accurate solvers and algorithms in CFD, the modified DGM presented in this thesis could be used to help and accelerate the adoption of high order methods in industry.

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Books on the topic "Weighted Discontinuous Galerkin method"

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Dolejší, Vít, and Miloslav Feistauer. Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3.

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Harold, Atkins, Keyes David, and Langley Research Center, eds. Parallel implementation of the discontinuous Galerkin method. National Aeronautics and Space Administration, Langley Research Center, 1999.

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Cockburn, B. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. ICASE, NASA Langley Research Center, 2000.

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Chi-Wang, Shu, and Institute for Computer Applications in Science and Engineering., eds. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. ICASE, NASA Langley Research Center, 2000.

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1967-, Ern Alexandre, ed. Mathematical aspects of discontinuous galerkin methods. Springer, 2012.

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United States. National Aeronautics and Space Administration., ed. An HP-adaptive discontinuous Galerkin method for hyperbolic conservation laws. The University of Texas at Austin ; [Washington, DC, 1994.

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United States. National Aeronautics and Space Administration., ed. An HP-adaptive discontinuous Galerkin method for hyperbolic conservation laws. The University of Texas at Austin ; [Washington, DC, 1994.

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United States. National Aeronautics and Space Administration., ed. An HP-adaptive discontinuous Galerkin method for hyperbolic conservation laws. The University of Texas at Austin ; [Washington, DC, 1994.

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Liu, Jianguo. A high order discontinuous Galerkin method for 2D incompressible flows. National Aeronautics and Space Administration, Langley Research Center, 1999.

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Cockburn, B. The Runge-Kutta discontinuous Galerkin method for convection-dominated problems. National Aeronautics and Space Administration, Langley Research Center, 2000.

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Book chapters on the topic "Weighted Discontinuous Galerkin method"

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Zunino, Paolo. "Mortar and Discontinuous Galerkin Methods Based on Weighted Interior Penalties." In Lecture Notes in Computational Science and Engineering. Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-75199-1_38.

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Dolejší, Vít, and Miloslav Feistauer. "Introduction." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_1.

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Dolejší, Vít, and Miloslav Feistauer. "Fluid-Structure Interaction." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_10.

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Dolejší, Vít, and Miloslav Feistauer. "DGM for Elliptic Problems." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_2.

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Dolejší, Vít, and Miloslav Feistauer. "Methods Based on a Mixed Formulation." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_3.

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Dolejší, Vít, and Miloslav Feistauer. "DGM for Convection-Diffusion Problems." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_4.

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Dolejší, Vít, and Miloslav Feistauer. "Space-Time Discretization by Multistep Methods." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_5.

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Dolejší, Vít, and Miloslav Feistauer. "Space-Time Discontinuous Galerkin Method." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_6.

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Dolejší, Vít, and Miloslav Feistauer. "Generalization of the DGM." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_7.

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Dolejší, Vít, and Miloslav Feistauer. "Inviscid Compressible Flow." In Discontinuous Galerkin Method. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-19267-3_8.

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Conference papers on the topic "Weighted Discontinuous Galerkin method"

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Xijun, He, Yang Dinghui, and Zhou Yanjie. "A weighted Runge-Kutta discontinuous Galerkin method for wavefield modeling." In SEG Technical Program Expanded Abstracts 2014. Society of Exploration Geophysicists, 2014. http://dx.doi.org/10.1190/segam2014-0579.1.

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Thompson, Lonny L. "Implementation of Non-Reflecting Boundaries in a Space-Time Finite Element Method for Structural Acoustics." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-3841.

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Abstract This paper examines the development and implementation of second-order accurate non-reflecting boundary conditions in a time-discontinuous Galerkin finite element method for structural acoustics in unbounded domains. The formulation is based on a multi-field space-time variational equation for both the acoustic fluid and elastic solid together with their interaction. This approach to the modeling of the temporal variables allows for the consistent use of high-order accurate adaptive solution strategies for unstructured finite elements in both time and space. An important feature of the method is the incorporation of temporal jump operators which allow for discretizations that are discontinuous in time. Two alternative approaches are examined for implementing non-reflecting boundaries within a time-discontinuous Galerkin finite element method; direct implementation of the exterior acoustic impedance through a weighted variational equation in time and space, and indirectly through a decomposition into two equations involving an auxiliary variable defined on the non-reflecting boundary. The idea for the indirect approach was originally developed in (Kallivokas, 1991) in the context of a standard semi-discrete formulation. Extensions to general convex boundaries are also discussed — numerical results are presented for acoustic scattering from an elongated structure using a first-order accurate boundary condition applied to an elliptical absorbing boundary.

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Peyret, Christophe, and Philippe Delorme. "Discontinuous Galerkin Method for Computational Aeroacoustics." In 12th AIAA/CEAS Aeroacoustics Conference (27th AIAA Aeroacoustics Conference). American Institute of Aeronautics and Astronautics, 2006. http://dx.doi.org/10.2514/6.2006-2568.

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Kim, Cheolwan, H. Chang, and Jang Yeon Lee. "Compact Higher-order Discontinuous Galerkin Method." In 11th AIAA/CEAS Aeroacoustics Conference. American Institute of Aeronautics and Astronautics, 2005. http://dx.doi.org/10.2514/6.2005-2824.

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Le Bouteiller, P., M. Ben Jemaa, H. Chauris, et al. "Discontinuous Galerkin Method for TTI Eikonal Equation." In 79th EAGE Conference and Exhibition 2017. EAGE Publications BV, 2017. http://dx.doi.org/10.3997/2214-4609.201701253.

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das Gupta, Arnob, and Subrata Roy. "Discontinuous Galerkin Method for Solving Magnetohydrodynamic Equations." In 53rd AIAA Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics, 2015. http://dx.doi.org/10.2514/6.2015-1616.

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Wukie, Nathan A., Paul D. Orkwis, and Christopher R. Schrock. "A Chimera-based, zonal discontinuous Galerkin method." In 23rd AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics, 2017. http://dx.doi.org/10.2514/6.2017-3947.

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Hirsch, Charles, Andrey Wolkov, and Benoit Leonard. "Discontinuous Galerkin Method on Unstructured Hexahedral Grids." In 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition. American Institute of Aeronautics and Astronautics, 2009. http://dx.doi.org/10.2514/6.2009-177.

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Peyret, Christophe, and Ph Delorme. "hp Discontinuous Galerkin Method for Computational Aeroacoustics." In 13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference). American Institute of Aeronautics and Astronautics, 2007. http://dx.doi.org/10.2514/6.2007-3475.

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Clément, J. B., F. Golay, M. Ersoy, and D. Sous. "Adaptive Discontinuous Galerkin Method for Richards Equation." In Topical Problems of Fluid Mechanics 2020. Institute of Thermomechanics, AS CR, v.v.i., 2020. http://dx.doi.org/10.14311/tpfm.2020.004.

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Reports on the topic "Weighted Discontinuous Galerkin method"

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Qiu, Jing-Mei, and Chi-Wang Shu. Convergence of High Order Finite Volume Weighted Essentially Non-Oscillatory Scheme and Discontinuous Galerkin Method for Nonconvex Conservation Laws. Defense Technical Information Center, 2007. http://dx.doi.org/10.21236/ada468107.

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Lin, Guang, and George E. Karniadakis. A Discontinuous Galerkin Method for Two-Temperature Plasmas. Defense Technical Information Center, 2005. http://dx.doi.org/10.21236/ada458981.

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Shu, Chi-Wang. Final Technical Report: High Order Discontinuous Galerkin Method and Applications. Office of Scientific and Technical Information (OSTI), 2019. http://dx.doi.org/10.2172/1499046.

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Greene, Patrick T., Samuel P. Schofield, and Robert Nourgaliev. Dynamic Mesh Adaptation for Front Evolution Using Discontinuous Galerkin Based Weighted Condition Number Mesh Relaxation. Office of Scientific and Technical Information (OSTI), 2016. http://dx.doi.org/10.2172/1260506.

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Romkes, A., S. Prudhomme, and J. T. Oden. A Posteriori Error Estimation for a New Stabilized Discontinuous Galerkin Method. Defense Technical Information Center, 2002. http://dx.doi.org/10.21236/ada438102.

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Nourgaliev, R., H. Luo, S. Schofield, et al. Fully-Implicit Orthogonal Reconstructed Discontinuous Petrov-Galerkin Method for Multiphysics Problems. Office of Scientific and Technical Information (OSTI), 2015. http://dx.doi.org/10.2172/1178386.

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Laeuter, Matthias, Francis X. Giraldo, Doerthe Handorf, and Klaus Dethloff. A Discontinuous Galerkin Method for the Shallow Water Equations in Spherical Triangular Coordinates. Defense Technical Information Center, 2007. http://dx.doi.org/10.21236/ada486030.

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Bui-Thanh, Tan, and Omar Ghattas. Analysis of an Hp-Non-conforming Discontinuous Galerkin Spectral Element Method for Wave. Defense Technical Information Center, 2011. http://dx.doi.org/10.21236/ada555327.

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Wang, Wei, Xiantao Li, and Chi-Wang Shu. The Discontinuous Galerkin Method for the Multiscale Modeling of Dynamics of Crystalline Solids. Defense Technical Information Center, 2007. http://dx.doi.org/10.21236/ada472151.

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Lieberman, Evan, Xiaodong Liu, Nathaniel Ray Morgan, Darby Jon Luscher, and Donald E. Burton. A higher-order Lagrangian discontinuous Galerkin hydrodynamic method for solid dynamics and reactive materials. Office of Scientific and Technical Information (OSTI), 2019. http://dx.doi.org/10.2172/1492638.

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