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39
The direct discontinuous Galerkin (ddg) methods for diffusion problems
 SIAM J. Numer. Anal
"... Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475698]. In this work, we show that higher orde ..."
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Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all pk elements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one and twodimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.
A Central Discontinuous Galerkin Method for HamiltonJacobi Equations
 J SCI COMPUT
, 2010
"... In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of HamiltonJacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin m ..."
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Cited by 15 (1 self)
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In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of HamiltonJacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since HamiltonJacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weightedresidual” or “stabilizationbased” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for HamiltonJacobi equations. The L2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general HamiltonJacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.
Local Discontinuous Galerkin Methods for HighOrder TimeDependent Partial Differential Equations
, 2010
"... Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, ..."
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Cited by 12 (1 self)
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Discontinuous Galerkin (DG) methods are a class of finite element methods using discontinuous basis functions, which are usually chosen as piecewise polynomials. Since the basis functions can be discontinuous, these methods have the flexibility which is not shared by typical finite element methods, such as the allowance of arbitrary triangulation with hanging nodes, less restriction in changing the polynomial degrees in each element independent of that in the neighbors (p adaptivity), and local data structure and the resulting high parallel efficiency. In this paper, we give a general review of the local DG (LDG) methods for solving highorder timedependent partial differential equations (PDEs). The important ingredient of the design of LDG schemes, namely the adequate choice of numerical fluxes, is highlighted. Some of the applications of the LDG methods for highorder timedependent PDEs are also be discussed.
Discontinuous Galerkin methods: general approach and stability, Numerical Solutions of Partial Differential Equations
 ADVANCED COURSES IN MATHEMATICS CRM BARCELONA, PAGES
, 2009
"... In these lectures, we will give a general introduction to the discontinuous Galerkin (DG) methods for solving time dependent, convection dominated partial differential equations (PDEs), including the hyperbolic conservation laws, convection diffusion equations, and PDEs containing higher order spati ..."
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Cited by 11 (2 self)
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In these lectures, we will give a general introduction to the discontinuous Galerkin (DG) methods for solving time dependent, convection dominated partial differential equations (PDEs), including the hyperbolic conservation laws, convection diffusion equations, and PDEs containing higher order spatial derivatives such as the KdV equations and other nonlinear dispersive wave equations. We will discuss cell entropy inequalities, nonlinear stability, and error estimates. The important ingredient of the design of DG schemes, namely the adequate choice of numerical fluxes, will be explained in detail. Issues related to the implementation of the DG method will also be addressed.
A Patchy Dynamic Programming Scheme for a Class of HamiltonJacobiBellman Equations
, 2011
"... In this paper we present a new parallel algorithm for the solution of HamiltonJacobiBellman equations related to optimal control problems. The main idea is to divide the domain of computation into subdomains following the dynamics of the control problem. This results in a rather complex geometric ..."
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Cited by 7 (1 self)
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In this paper we present a new parallel algorithm for the solution of HamiltonJacobiBellman equations related to optimal control problems. The main idea is to divide the domain of computation into subdomains following the dynamics of the control problem. This results in a rather complex geometrical subdivision, but has the advantage that every subdomain is invariant with respect to the optimal controlled vector field, so that we can compute the value function in each subdomain assigning the task to a processor and avoiding the classical transmission condition on the boundaries of the subdomains. For this specific feature the subdomains are patches in the sense introduced by Ancona and Bressan in [1]. Several examples in dimension two and three illustrate the properties of the new method.
A second order discontinuous Galerkin fast sweeping method for eikonal equations
 Journal of Computational Physics
, 2008
"... ABSTRACT In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin ..."
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ABSTRACT In this paper, we construct a second order fast sweeping method with a discontinuous Galerkin
A DISCONTINUOUS GALERKIN SCHEME FOR FRONT PROPAGATION WITH OBSTACLES
"... Abstract. We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu [6]), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et ..."
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Abstract. We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu [6]), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. [8], leading to a level set formulation driven by min(ut+H(x,∇u),u−g(x)) = 0, where g(x) is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian H is a linear function of∇u, correspondingtolinear convectionproblems in presence ofobstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a RungeKutta third order time discretization using the technique proposed in Zhang and Shu [22]. Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost. 1.
A discontinuous Galerkin solver for front propagation
, 2009
"... Abstract. We propose a new discontinuous Galerkin (DG) method based on [9] to solve a class of HamiltonJacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical expe ..."
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Cited by 5 (3 self)
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Abstract. We propose a new discontinuous Galerkin (DG) method based on [9] to solve a class of HamiltonJacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation. The HamiltonJacobi (HJ) equation 1.
Uniformly accurate discontinuous Galerkin fast sweeping methods for Eikonal equations
 SIAM J. on Sci. Comp
, 2011
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Optimal Control of the Classical TwoPhase Stefan Problem in Level Set Formulation
, 2010
"... Optimal control (motion planning) of the free interface in classical twophase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The firstorder optimality system is derived on a formal basis. It provides gradient information based on the adjoint ..."
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Cited by 4 (0 self)
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Optimal control (motion planning) of the free interface in classical twophase Stefan problems is considered. The evolution of the free interface is modeled by a level set function. The firstorder optimality system is derived on a formal basis. It provides gradient information based on the adjoint temperature and adjoint level set function. Suitable discretization schemes for the forward and adjoint systems are described. Numerical