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Chi-wang Shu - One of the best experts on this subject based on the ideXlab platform.
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runge kutta discontinuous Galerkin Method using weno limiters ii
Journal of Computational Physics, 2008Co-Authors: Jun Zhu, Chi-wang Shu, Jianxian Qiu, Michael DumbserAbstract:In J. Qiu, C.-W. Shu, Runge-Kutta discontinuous Galerkin Method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907-929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume Methodology as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) Methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the Method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG Methods. Numerical results are provided to illustrate the behavior of this procedure.
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Reinterpretation and simplified implementation of a discontinuous Galerkin Method for Hamilton-jacobi equations
Applied Mathematics Letters, 2005Co-Authors: Chi-wang ShuAbstract:Abstract In this note, we reinterpret a discontinuous Galerkin Method originally developed by Hu and Shu [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element Method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690] (see also [O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin Method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434]) for solving Hamilton–Jacobi equations. With this reinterpretation, numerical solutions will automatically satisfy the curl-free property of the exact solutions inside each element. This new reinterpretation allows a Method of lines formulation, which renders a more natural framework for stability analysis. Moreover, this reinterpretation renders a significantly simplified implementation with reduced cost, as only a smaller subspace of the original solution space in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element Method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin Method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is used and the least squares procedure used in [C. Hu, C.-W. Shu, A discontinuous Galerkin finite element Method for Hamilton–Jacobi equations, SIAM Journal on Scientific Computing 21 (1999) 666–690; O. Lepsky, C. Hu, C.-W. Shu, Analysis of the discontinuous Galerkin Method for Hamilton–Jacobi equations, Applied Numerical Mathematics 33 (2000) 423–434] is completely avoided.
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a local discontinuous Galerkin Method for kdv type equations
2001Co-Authors: Jue Yan, Chi-wang ShuAbstract:In this paper we develop a local discontinuous Galerkin Method for solving KdV type equations containing third derivative terms in one and two space dimensions. The Method is based on the framework of the discontinuous Galerkin Method for conservation laws and the local discontinuous Galerkin Method for viscous equations containing second derivatives; however, the guiding principle for intercell fluxes and nonlinear stability is new. We prove L2 stability and a cell entropy inequality for the square entropy for a class of nonlinear PDEs of this type in both one and multiple space dimensions, and we give an error estimate for the linear cases in the one-dimensional case. The stability result holds in the limit case when the coefficients to the third derivative terms vanish; hence the Method is especially suitable for problems which are "convection dominated," i.e., those with small second and third derivative terms. Numerical examples are shown to illustrate the capability of this Method. The Method has the usual advantage of local discontinuous Galerkin Methods, namely, it is extremely local and hence efficient for parallel implementations and easy for h-p adaptivity.
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the runge kutta discontinuous Galerkin Method for conservation laws v
Journal of Computational Physics, 1998Co-Authors: Bernardo Cockburn, Chi-wang ShuAbstract:This is the fifth paper in a series in which we construct and study the so-called Runge?Kutta discontinuous Galerkin Method for numerically solving hyperbolic conservation laws. In this paper, we extend the Method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two-dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles on the quality of the approximation.
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Quadrature-Free Implementation of Discontinuous Galerkin Method for Hyperbolic Equations
AIAA Journal, 1998Co-Authors: Harold L. Atkins, Chi-wang ShuAbstract:A discontinuous Galerkin formulation that avoids the use of discrete quadrature formulas is described and applied to linear and nonlinear test problems in one and two space dimensions. This approach requires less computational time and storage than conventional implementations but preserves the compactness and robustness inherent in the discontinuous Galerkin Method. Test problems include the linear and nonlinear one-dimensional scalar advection of smooth initial value problems that are discretized by using unstructured grids with varying degrees of smoothness and regularity, and two-dimensional linear Euler solutions on unstructured grids.
Mehdi Dehghan - One of the best experts on this subject based on the ideXlab platform.
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local discontinuous Galerkin Method for distributed order time fractional diffusion wave equation application of laplace transform
Authorea Preprints, 2020Co-Authors: Hadi Mohammadi Firouzjaei, Hojatollah Adibi, Mehdi DehghanAbstract:In this paper, the Laplace transform combined with the local discontinuous Galerkin Method is used for distributed-order time-fractional diffusion-wave equation. In this Method,at first, we convert the equation to some time-independent problems by Laplace transform.Then we can solve these stationary equations by the local discontinuous Galerkin Method to discretize diffusion operators at the same time. Then, by using a numerical inversion of the Laplace transform we can find the solutions of the original equation. One of the advantages of this procedure is its parallel implementation. Another advantage of this approach is that the number of stationary problems that should be solved is much less than that are needed in time-marching Methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the Method.
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direct meshless local petrov Galerkin Method for elliptic interface problems with applications in electrostatic and elastostatic
Computer Methods in Applied Mechanics and Engineering, 2014Co-Authors: Ameneh Taleei, Mehdi DehghanAbstract:Abstract In recent years, there have been extensive efforts to find the numerical Methods for solving problems with interface. The main idea of this work is to introduce an efficient truly meshless Method based on the weak form for interface problems. The proposed Method combines the direct meshless local Petrov–Galerkin Method with the visibility criterion technique to solve the interface problems. It is well-known in the classical meshless local Petrov–Galerkin Method, the numerical integration of local weak form based on the moving least squares shape function is computationally expensive. The direct meshless local Petrov–Galerkin Method is a newly developed modification of the meshless local Petrov–Galerkin Method that any linear functional of moving least squares approximation will be only done on its basis functions. It is done by using a generalized moving least squares approximation, when approximating a test functional in terms of nodes without employing shape functions. The direct meshless local Petrov–Galerkin Method can be a very attractive scheme for computer modeling and simulation of problems in engineering and sciences, as it significantly uses less computational time in comparison with the classical meshless local Petrov–Galerkin Method. To create the appropriate generalized moving least squares approximation in the vicinity of an interface, we choose the visibility criterion technique that modifies the support of the weight (or kernel) function. This technique, by truncating the support of the weight function, ignores the nodes on the other side of the interface and leads to a simple and efficient computational procedure for the solution of closed interface problems. In the proposed Method, the essential boundary conditions and the jump conditions are directly imposed by substituting the corresponding terms in the system of equations. Also, the Heaviside step function is applied as the test function in the weak form on the local subdomains. Some numerical tests are given including weak and strong discontinuities in the Poisson interface problem. To demonstrate the application of these problems, linearized Poisson–Boltzmann and linear elasticity problems with two phases are studied. The proposed Method is compared with analytical solution and the meshless local Petrov–Galerkin Method on several test problems taken from the literature. The numerical results confirm the effectiveness of the proposed Method for the interface problems and also provide significant savings in computational time rather than the classical meshless local Petrov–Galerkin Method.
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ritz Galerkin Method with bernstein polynomial basis for finding the product solution form of heat equation with non classic boundary conditions
International Journal of Numerical Methods for Heat & Fluid Flow, 2012Co-Authors: S A Yousefi, Zahra Barikbin, Mehdi DehghanAbstract:Purpose – The purpose of this paper is to implement the Ritz‐Galerkin Method in Bernstein polynomial basis to give approximation solution of a parabolic partial differential equation with non‐local boundary conditions.Design/Methodology/approach – The properties of Bernstein polynomial and Ritz‐Galerkin Method are first presented, then the Ritz‐Galerkin Method is utilized to reduce the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.Findings – The authors applied the Method presented in this paper and solved three test problems.Originality/value – This is the first time that the Ritz‐Galerkin Method in Bernstein polynomial basis is employed to solve the model investigated in the current paper.
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bernstein ritz Galerkin Method for solving an initial boundary value problem that combines neumann and integral condition for the wave equation
Numerical Methods for Partial Differential Equations, 2009Co-Authors: S A Yousefi, Zahra Barikbin, Mehdi DehghanAbstract:In this article, the Ritz-Galerkin Method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz-Galerkin Method are first presented, then Ritz-Galerkin Method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010
Jan S Hesthaven - One of the best experts on this subject based on the ideXlab platform.
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discontinuous Galerkin Method for fractional convection diffusion equations
SIAM Journal on Numerical Analysis, 2014Co-Authors: Jan S HesthavenAbstract:We propose a discontinuous Galerkin Method for fractional convection-diffusion equations with a superdiffusion operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and a fractional integral of order $2-\alpha$. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin Method scheme is proposed for the equations. We prove stability and optimal order of convergence ${\cal O}(h^{k+1})$ for the fractional diffusion problem, and an order of convergence of ${\cal O}(h^{k+\frac{1}{2}})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.
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analysis and application of the nodal discontinuous Galerkin Method for wave propagation in metamaterials
Journal of Computational Physics, 2014Co-Authors: Jan S HesthavenAbstract:In this paper, we develop a nodal discontinuous Galerkin Method for solving the time-dependent Maxwell?s equations when metamaterials are involved. Both semi- and fully-discrete schemes are constructed. Numerical stability and error estimate are proved for both schemes. Numerical results are presented to demonstrate that the Method is not only efficient but also very effective in solving metamaterial Maxwell?s equations.
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discontinuous Galerkin Method for fractional convection diffusion equations
arXiv: Numerical Analysis, 2013Co-Authors: Jan S HesthavenAbstract:We propose a discontinuous Galerkin Method for convection-subdiffusion equations with a fractional operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and fractional integrals of order $2-\alpha$, and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin Method scheme is derived for the equations. We prove stability and optimal order of convergence O($h^{k+1}$) for subdiffusion, and an order of convergence of ${\cal O}(h^{k+1/2})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.
Mats G Larson - One of the best experts on this subject based on the ideXlab platform.
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a continuous discontinuous Galerkin Method and a priori error estimates for the biharmonic problem on surfaces
IEEE Communications Magazine, 2017Co-Authors: Karl Larsson, Mats G LarsonAbstract:We present a continuous/discontinuous Galerkin Method for approximating solutions to a fourth order elliptic PDE on a surface embedded in R-3. A priori error estimates, taking both the approximatio ...
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the continuous Galerkin Method is locally conservative
Journal of Computational Physics, 2000Co-Authors: Thomas J R Hughes, Gerald Engel, Luca Mazzei, Mats G LarsonAbstract:We examine the conservation law structure of the continuous Galerkin Method. We employ the scalar, advection?diffusion equation as a model problem for this purpose, but our results are quite general and apply to time-dependent, nonlinear systems as well. In addition to global conservation laws, we establish local conservation laws which pertain to subdomains consisting of a union of elements as well as individual elements. These results are somewhat surprising and contradict the widely held opinion that the continuous Galerkin Method is not locally conservative.
Bernardo Cockburn - One of the best experts on this subject based on the ideXlab platform.
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a hybridizable discontinuous Galerkin Method for steady state convection diffusion reaction problems
SIAM Journal on Scientific Computing, 2009Co-Authors: Bernardo Cockburn, Bo Dong, Johnny Guzman, Marco Restelli, Riccardo SaccoAbstract:In this article, we propose a novel discontinuous Galerkin Method for convection-diffusion-reaction problems, characterized by three main properties. The first is that the Method is hybridizable; t...
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the runge kutta discontinuous Galerkin Method for conservation laws v
Journal of Computational Physics, 1998Co-Authors: Bernardo Cockburn, Chi-wang ShuAbstract:This is the fifth paper in a series in which we construct and study the so-called Runge?Kutta discontinuous Galerkin Method for numerically solving hyperbolic conservation laws. In this paper, we extend the Method to multidimensional nonlinear systems of conservation laws. The algorithms are described and discussed, including algorithm formulation and practical implementation issues such as the numerical fluxes, quadrature rules, degrees of freedom, and the slope limiters, both in the triangular and the rectangular element cases. Numerical experiments for two-dimensional Euler equations of compressible gas dynamics are presented that show the effect of the (formal) order of accuracy and the use of triangles or rectangles on the quality of the approximation.
