A local discontinuous galerkin method for kdv type. Local discontinuous galerkin method for solving burgers. In this work, we design, analyze and test a local discontinuous galerkin method for solving the burgers poisson equation. The resulting scheme becomes vk h d r u k h, m k du k h dt. Discontinuous galerkin dg method dg method for burgers equation cont. Numerical schemes applied to the burgers and buckleyleverett. In this present work discontinuous galerkin finite element method is used for 1d inviscid burgers equation. Discontinuous galerkin method for burgers equation m. Onedimensional equations drawbacks such as implementation complexity, residue convergence problems and, normally, lack of subcell resolution vincent and jameson, 2011. Three standard problems are used to validate the proposed algorithm. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. A staggered discontinuous galerkin method with local tv regularization for the burgers equation hiu ning chan and eric t. Numerical solution of the burgers equation by local.
Numerical schemes applied to the burgers and buckley. A local discontinuous galerkin method for the burgerspoisson. The temporal discretization is accomplished by the total variation diminishing rungekutta method. While these methods have been known since the early 1970s, they have experienced a phenomenal growth in interest dur. Poissons equation and helmholtz equation are elliptic equations, the linear advection and burgers equations are hyperbolic equations, and the advection di. A new discontinuous petrovgalerkin method with optimal test. Discontinuous galerkin solutions of the navierstokes. Compact and stable discontinuous galerkin methods for. A staggered discontinuous galerkin method with local tv. The regionallyimplicit discontinuous galerkin method.
Burgers and coupled burgers equations are solved by the local discontinuous galerkin method. Local discontinuous galerkin method for solving burgers and coupled burgers equations article pdf available in chinese physics b 2011 november 2011 with 104 reads how we measure reads. Adaptive tvdrk discontinuous galerkin algorithms for shallow. Discontinuous galerkin method for fractional convection. In this work, we analyze and test a local discontinuous galerkin method for solving the burgerstype equation. The discontinuous galerkin method for conservation. Based on the previous discussions of the discontinuous galerkin methods, it is tempting to simply write the heat equation as. Discontinuous galerkin finite element method for the nonlinear hyperbolic problems with entropybased artificial viscosity stabilization a dissertation by valentin nikolaevich zingan. In this paper we consider the advection equation, the burgers equation, and the. Numerical algorithm based on an implicit fully discrete local.
Vanishing viscosity and local discontinuous galerkin methods zhiping maoa, george em karniadakisa. Numerical simulations with the galerkin least squares finite element method for the burgers equation on the real line in this work we present an efficient galerkin least squares finite element scheme to simulate the burgers equation on the whole real line and subjected to. The hdpg method stems from a combination of the hybridized discontinuous galerkin hdg method and the theory of the optimal test functions, suitably modified to enforce the conservativity at the element level. Request pdf the local discontinuous galerkin finite element method for burgers equation in this paper, we study the local discontinuous galerkin ldg finite. Abstract we consider the viscous burgers equation with a fractional nonlinear term as a model involving nonlocal. A hybridized discontinuous petrovgalerkin scheme for. The staggered discontinuous galerkin sdg method has been recently developed for the numerical approximation of partial differential equations. A discontinuous galerkin method for stochastic conservation laws yunzhang li, chiwang shu y, shanjian tangz abstract in this paper we present a discontinuous galerkin dg method to approximate stochastic conservation laws, which is an e cient highorder scheme.
Discontinuous galerkin methods lecture 8 brown university. In the current work, we extend the local discontinuous galerkin method to a more general application system. Darmofal aerospace computational design laboratory, massachusetts insititute of technology a newtonkrylov method is developed for the solution of the steady compressible navierstokes equations using a discontinuous galerkin dg discretization on unstructured. We describe the algorithm formulation and practical implementation of the local discontinuous galerkin method in detail. Numerical methods for hyperbolic conservation laws 9 6. Numerical algorithm based on an implicit fully discrete. The numerical results indicate that the method is very accurate and efficient.
A discontinuous galerkin method for stochastic conservation laws. Development of discontinuous galerkin method for 1d. We prove that the scheme is unconditionally stable and convergent. Direct discontinuous galerkin method for the generalized burgersfisher equation article pdf available in chinese physics b 219. High and low reynolds numbers play important roles in both modeling and numerical simulation. Fractional burgers equation with nonlinear nonlocality. Spectral vanishing viscosity and local discontinuous galerkin methods zhiping maoa, george em karniadakisa.
Pdf local discontinuous galerkin method for solving burgers and. The proposed numerical method is high order accurate and preserves two. The comparison is performed using as much shared code as possible between the two. To show the shape of burgers equation introducing the. Notes on burgerss equation maria cameron contents 1. Tvb rungekutta local projection discontinuous galerkin.
The key feature of the proposed method is formed by combining the. Subject terms diffusion equation, poisson equation, advectiondiffusion equation, burgers equation, discontinuous galerkin methods, recovery method, pane. Adaptive tvdrk discontinuous galerkin algorithms for. Compact and stable discontinuous galerkin methods for convectiondiffusion problems s. In this paper, the burgers equation is transformed into the linear diffusion equation by using the hopfcole transformation. The numerical results indicate that the method is very accurate. Burgers equation frequently appears in the study of turbulence theory, as well as some other scientific fields. We consider the solution of the viscous burgers equation. Numerical simulations with the galerkin least squares finite. Local discontinuous galerkin methods for three classes of nonlinear wave equations 251 where f u, a u.
The method is applied to the solution of the onedimensional viscous burgers equation and two forms of the modi. Discontinuous galerkin method for hyperbolic conservation laws ioanna mousikou hyperbolic conservation laws form a special class of partial di. It is desirable to design high order mass preserving numerical schemes for solving this equation. Numerical approximation of timefractional burgerstype.
The hybridized discontinuous petrovgalerkin scheme hdpg for compressible flows is presented. In 1d, a formulation was successfully implemented to solve continuous problems. This scheme is applied to 1d nonlinear conservation equation burgers equation in which the governing. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal. In this work, we analyze and test a local discontinuous galerkin method for solving the burgers type equation. A staggered discontinuous galerkin method with local tv regularization for the burgers equation volume 8 issue 4 hiu ning chan, eric t. Roberts the institute for computational engineering and sciences the university of texas at austin austin, texas 78712 by ices report 1025 june 2010. They describe phenomena that involve conserved quantities and their solutions show discontinuities which re. Discontinuous problems for the in viscid burgers equation and a breaking dam problem were also investigated. A local discontinuous galerkin method for the burgerspoisson equation hailiang liu and nattapol ploymaklam abstract. This model, proposed by whitham linear and nonlinear waves, 1974 as a simplified model for shallow water waves, admits conservation of both momentum and energy as two invariants. Development of discontinuous galerkin method for 1d inviscid. Implementation of the entropy viscosity method with the discontinuous galerkin.
A local discontinuous galerkin method for kdv type equations. Recoverybased discontinuous galerkin for navierstokes. A spacetime discontinuous galerkin spectral element. The discontinuous galerkin method considered in 20 refers to a discontinuous galerkin discretization in time, hence is di. Consider hyperbolic conservation laws of the form 2 q. Roberts the institute for computational engineering and sciences the university of texas at austin. Numerical experiments are given to verify the efficiency and accuracy of our method. The discontinuous galerkin method for conservation laws. The proposed numerical method, which is highorder accurate, is based on a finite difference scheme in time and local discontinuous galerkin methods in space. Numerical approximation of timefractional burgerstype equation. Local discontinuous galerkin methods for three classes of nonlinear wave equations 251 where fu, au. Numerical simulations with the galerkin least squares.
Viscous shock capturing in a timeexplicit discontinuous galerkin. Burgers equation, and so one needs to help a littlefor example by adding an artificial. The discontinuous basis functions in dg worked well for advection. Though very simple, this equation can be regarded as a model for decaying free turbulence. While these methods have been known since the early 1970s, they have experienced a.
Numerical experiments are given to verify the efficiency and. In 1d, a formulation was successfully implemented to. The main objective of this research work is to apply the discontinuous galerkin method to a classical partial differential equation to investigate the properties of the numerical solution and compare the numerical solution to the analytical solution by using discontinuous galerkin method. Discontinuous galerkin dg methods 16, 31, 43, 48 are a combination of ideas from finite. Because of the linearity, the solution can be obtained essentially explicitly by following the characteristic directions, hence avoiding the abovementioned. Numerical simulations with the galerkin least squares finite element method for the burgers equation on the real line in this work we present an efficient galerkin least squares finite element scheme to simulate the burgers equation on the whole real line and subjected to initial conditions with compact support. Adaptive tvdrk discontinuous galerkin algorithms for shallow water equations thida pongsanguansin, khamron mekchay and montri maleewong abstract the adaptive discontinuous galerkin dg method for solving the onedimensional conservation equation is presented. The schemes we present extend the previous work of yan and shu 30, 31 and of levy, shu. In this case we compare the e ciency and accuracy of our method against the fourthorder rungekutta discontinous galerkin rkdg scheme.
The burgers and coupled burgers equations are solved by the local discontinuous galerkin method. Wkhorfdo the discontinuous petrov galerkin euler equations in. Pdf local discontinuous galerkin method for solving. In this paper, we apply a numerical scheme to solve a twodimensional timefractional burgers equation. A new discontinuous petrovgalerkin method with optimal test functions. On positivitypreserving high order discontinuous galerkin schemes for compressible euler equations on rectangular meshes. A galerkin solution for burgers equation using cubic b. Lastly, we compare the various rdg schemes by fourier analysis. Solution of the burgers equation with nonzero viscosity 1 2. The local discontinuous galerkin finite element method for burgers. The discontinuous galerkin method for conservation laws october 14, 2010 the aim of this project is to study discontinuous galerkin methods applied to coupled systems of partial di erential equations in conservative form in 1d and 2d.
Local discontinuous galerkin methods for nonlinear schro. Some numerical tests are provided to illustrate the. In this paper, an implicit fully discrete local discontinuous galerkin ldg finite element method is considered for solving the time. Mass preserving discontinuous galerkin methods for. Brief history of recoverybased discontinuous galerkin reed and hill1 introduced discontinuous galerkin dg for neutron masstransport equations in 1973. Spectral vanishing viscosity and local discontinuous galerkin methods. Solution of 1d burgers and navierstokes equations j. On the comparison of the finite volume and discontinuous galerkin methods corrected version katja baumbach.
Modified burgers equation by the local discontinuous galerkin method. A new discontinuous petrovgalerkin method with optimal. In 18, lesaint and raviart first introduced the discontinuous galerkin finite element method for solving the neutron transport equation, which is a linear version of 1. A spacetime discontinuous galerkin spectral element method is combined with two different approaches for treating problems with discontinuous solutions.
On the comparison of the finite volume and discontinuous. Diffusionbased limiters for discontinuous galerkin methods part i. In this paper, we develop a class of direct discontinuous galerkin ddg methods for 1. Discontinuous galerkin method for hyperbolic conservation laws thesis by ioanna mousikou. Discontinuous galerkin method for hyperbolic conservation.
Nodal discontinuous galerkin methods algorithms, analysis, and applications this book discusses the discontinuous galerkin family of computational methods for solving partial differential equations. Request pdf development of discontinuous galerkin method for 1d inviscid burgers equation the main objective of this research work is to apply the discontinuous galerkin method to a classical. A local discontinuous galerkin method for the burgers. Hermite weno schemes and their application as limiters for. Department of mathematics, the chinese university of hong kong, sha tin, hong kong. Discontinuous galerkin method for hyperbolic conservation laws. Because of the linearity, the solution can be obtained essentially explicitly by following the characteristic directions. The obtained linear diffusion equation is discretized in space by the local discontinuous galerkin method. Equations using linear multigrid preconditioning laslo t. Accepted in revised version 28 april 2014 abstract.
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