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Solving elastic wave equations in 2D transversely isotropic media by a weighted Runge-Kutta discontinuous galerkin method

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Petroleum Science 2023年第2期20卷 827-839页

作者： Xi-Jun He Jing-Shuang Li Xue-Yuan Huang Yan-Jie Zhou School of Mathematics and Statistics Beijing Technology and Business University(BTBU)Beijing100048China School of Science China University of Mining and Technology(Beijing)Beijing100083China

Accurate wave propagation simulation in anisotropic media is important for forward modeling, migration and inversion. In this study, the weighted Runge-Kutta discontinuous galerkin (RKDG) method is extended to solve the elastic wave equations in 2D transversely isotropic media. The spatial discretization is based on the numerical flux discontinuous galerkin scheme. An explicit weighted two-step iterative Runge-Kutta method is used as time-stepping algorithm. The weighted RKDG method has good flexibility and applicability of dealing with undulating geometries and boundary conditions. To verify the correctness and effectiveness of this method, several numerical examples are presented for elastic wave propagations in vertical transversely isotropic and tilted transversely isotropic media. The results show that the weighted RKDG method is promising for solving wave propagation problems in complex anisotropic medium.

关键词： discontinuous galerkin method Anisotropy Transversely isotropic Modeling

Goal-Oriented Anisotropic hp-Adaptive discontinuous galerkin method for the Euler Equations

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Communications on Applied Mathematics and Computation 2022年第1期4卷 143-179页

作者： Vít Dolejší Filip Roskovec Faculty of Mathematics and Physics Charles UniversitySokolovská8318675 PragueCzech Republic

We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous galerkin(DG)method with focus on the goal-oriented error estimates and adaptivity.We analyse the adjoint consistency of the DG scheme where the adjoint problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms.Furthermore,we present the goal-oriented anisotropic hp-mesh adaptation method for the Euler equations.The theoretical results are supported by numerical experiments.

关键词： Euler equations discontinuous galerkin method Target functional Adjoint consistency Anisotropic hp-mesh adaptation

Conservative discontinuous galerkin/Hermite Spectral method for the Vlasov-Poisson System

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Communications on Applied Mathematics and Computation 2022年第1期4卷 34-59页

作者： Francis Filbet Tao Xiong Institut de Mathématiques de Toulouse UniversitéPaul SabatierToulouse 31062France School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamen 361005Fujian ProvinceChina

We propose a class of conservative discontinuous galerkin methods for the Vlasov-Pois-son system written as a hyperbolic system using Hermite polynomials in the velocity vari-able.These schemes are designed to be systematically as accurate as one wants with prov-able conservation of mass and possibly total energy.Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system.The proposed scheme employs the discontinuous galerkin discretization for both the Vlasov and the Poisson equations,resulting in a consistent description of the distribu-tion function and the electric field.Numerical simulations are performed to verify the order of the accuracy and conservation properties.

关键词： Energy conserving discontinuous galerkin method Hermite spectral method Vlasov-Poisson

A DEEP LEARNING BASED discontinuous galerkin method FOR HYPERBOLIC EQUATIONS WITH discontinuous SOLUTIONS AND RANDOM UNCERTAINTIES

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Journal of Computational Mathematics 2023年第6期41卷 1281-1304页

作者： Jingrun Chen Shi Jin Liyao Lyu School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026China Suzhou Institute for Advanced Research University of Science and Technology of ChinaSuzhou 215123China School of Mathematical Sciences Institute of Natural Sciencesand MOE-LSCShanghai Jiao Tong UniversityShanghai200240China Department of Computational Mathematics Scienceand EngineeringMichigan State UniversityEast LansingMI48824USA

We propose a deep learning based discontinuous galerkin method(D2GM)to solve hyperbolic equations with discontinuous solutions and random uncertainties.The main computational challenges for such problems include discontinuities of the solutions and the curse of dimensionality due to uncertainties.Deep learning techniques have been favored for high-dimensional problems but face difficulties when the solution is not smooth,thus have so far been mainly used for viscous hyperbolic system that admits only smooth solutions.We alleviate this difficulty by setting up the loss function using discrete shock capturing schemes–the discontinous galerkin method as an example–since the solutions are smooth in the discrete space.The convergence of D2GM is established via the Lax equivalence theorem kind of argument.The high-dimensional random space is handled by the Monte-Carlo method.Such a setup makes the D2GM approximate high-dimensional functions over the random space with satisfactory accuracy at reasonable cost.The D2GM is found numerically to be first-order and second-order accurate for(stochastic)linear conservation law with smooth solutions using piecewise constant and piecewise linear basis functions,respectively.Numerous examples are given to verify the efficiency and the robustness of D2GM with the dimensionality of random variables up to 200 for(stochastic)linear conservation law and(stochastic)Burgers’equation.

关键词： discontinuous galerkin method Loss function Convergence analysis Deep learning Hyperbolic equation

A Well-Balanced Runge-Kutta discontinuous galerkin method for Multilayer ShallowWater Equations with Non-Flat Bottom Topography

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Advances in Applied Mathematics and Mechanics 2022年第3期14卷 725-758页

作者： Nouh Izem Mohammed Seaid Laboratory of Engineering Sciences LabSI Faculty of ScienceIbn Zohr University AgadirMorocco Department of Engineering University of DurhamSouth RoadDurham DH13LEUK InternationalWater Research Institute University Mohammed VI PolytechnicBenguerirMorocco

A well-balanced Runge-Kutta discontinuous galerkin method is presented for the numerical solution of multilayer shallow water equations with mass exchange and non-ﬂat bottom topography.The governing equations are reformulated as a non-linear system of conservation laws with differential source forces and reaction terms.Coupling between theﬂow layers is accounted for in the system using a set of ex-change relations.The considered well-balanced Runge-Kutta discontinuous galerkin method is a locally conservativeﬁnite element method whose approximate solutions are discontinuous across the inter-element boundaries.The well-balanced property is achieved using a special discretization of source terms that depends on the nature of hydrostatic solutions along with the Gauss-Lobatto-Legendre nodes for the quadra-ture used in the approximation of source terms.The method can also be viewed as a high-order version of upwindﬁnite volume solvers and it offers attractive features for the numerical solution of conservation laws for which standardﬁnite element methods fail.To deal with the source terms we also implement a high-order splitting operator for the time integration.The accuracy of the proposed Runge-Kutta discontinuous galerkin method is examined for several examples of multilayer free-surfaceﬂows over bothﬂat and non-ﬂat beds.The performance of the method is also demonstrated by comparing the results obtained using the proposed method to those obtained using the incompressible hydrostatic Navier-Stokes equations and a well-established kinetic method.The proposed method is also applied to solve a recirculationﬂow problem in the Strait of Gibraltar.

关键词： discontinuous galerkin method well-balanced discretization Runge-Kutta scheme multilayer shallow water equations free-surfaceﬂows mass exchange wind-drivenﬂows strait of Gibraltar

Uniform Convergence Analysis of the discontinuous galerkin method on Layer-Adapted Meshes for Singularly Perturbed Problem

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Wuhan University Journal of Natural Sciences 2023年第5期28卷 411-420页

作者： SHI Jiamin LU Zhongshu ZHANG Luyi LU Sunjia CHENG Yao School of Mathematical Sciences Suzhou University of Science and TechnologySuzhou 215009JiangsuChina

This paper concerns a discontinuous galerkin(DG)method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations.We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework.The convergence rate is valid independent of the small parameter.Furthermore,we establish a sharper L^(2)-error estimate if the true solution has a special regular component.Numerical experiments are also given.

关键词： layer-adapted meshes singularly perturbed problem uniform convergence discontinuous galerkin method

Hermite WENO-based limiters for high order discontinuous galerkin method on unstructured grids

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Acta Mechanica Sinica 2012年第2期28卷 241-252页

作者： Zhen-Hua Jiang Chao Yan Jian Yu Wu Yuan College of Aeronautics Science and Engineering Beihang University100191 BeijingChina

A novel class of weighted essentially nonoscillatory （WENO） schemes based on Hermite polynomi- als, termed as HWENO schemes, is developed and applied as limiters for high order discontinuous galerkin （DG） method on triangular grids. The developed HWENO methodology utilizes high-order derivative information to keep WENO re- construction stencils in the von Neumann neighborhood. A simple and efficient technique is also proposed to enhance the smoothness of the existing stencils, making higher-order scheme stable and simplifying the reconstruction process at the same time. The resulting HWENO-based limiters are as compact as the underlying DG schemes and therefore easy to implement. Numerical results for a wide range of flow conditions demonstrate that for DG schemes of up to fourth order of accuracy, the designed HWENO limiters can simul- taneously obtain uniform high order accuracy and sharp, es- sentially non-oscillatory shock transition.

关键词： discontinuous galerkin method Limiters WENO. High order accuracy. Unstructured grids

FINITE ELEMENT AND discontinuous galerkin method FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO-AND THREE-DIMENSIONS

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Journal of Computational Mathematics 2008年第5期26卷 702-715页

作者： Yanzhao Cao Ran Zhang Kai Zhang Department of Mathematics Florida A&M University TaUahassee FL32307 USA Department of Mathematics Jilin University Changchun 130023 China

In this paper, we consider the finite element method and discontinuous galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.

关键词： Stochastic partial differential equation Finite element method discontinuous galerkin method Stochastic Helmholtz equation.

Effect of channel shape on selection of time marching scheme in the discontinuous galerkin method for 1-D open channel flow

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Journal of Hydrodynamics 2015年第3期27卷 413-426页

作者： SAFARZADEH MALEKI Farzam KHAN Abdul A. Engineering Department Massachusetts Maritime Academy Glenn Department of Civil Engineering Clemson University

One-dimensional open channel flows are simulated using the discontinuous galerkin finite element method. Three different explicit time marching schemes, including multistep/multistage schemes, are evaluated for different channel shapes for accuracy and efficiency. The Forward Euler, second-order Adam-Bashforth （multistep）, and second-order total variation diminishing （TVD） Runge-Kutta （multistage） time marching schemes are utilized. The role of monotonized central, minmod, and zero TVD slope limiters for each of the time marching scheme is investigated. The numerical flux is approximated using HLL function. The accuracy and robustness of different time marching schemes are evaluated for steady and unsteady flows using analytical and measured data. The unsteady flows include dam break tests with wet and dry beds downstream of the dam in prismatic （rectangular, trapezoidal, triangular, and parabolic cross-sections） and non-prismatic （natural river） channels. The steady flow test involves simulation of hydraulic jump in a diverging rectangular channel. The various schemes are evaluated by comparing accuracy using statistical measures and efficiency using maximum possible time step size as well as CPU runtime. The second-order Adam-Bashforth time marching scheme is found to have the best accuracy and efficiency among the time stepping schemes tested.

关键词： discontinuous galerkin method shallow water equations time marching schemes total variation diminishing （TVD）slope limiter