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In this paper, we propose the local discontinuous Galerkin method based on the generalized alternating numerical flux for solving the one-dimensional second-order wave equation with the periodic boundary conditions. Introducing two auxiliary variables, the second-order equation is rewritten into the first-order equation systems. We prove the stability and energy conservation of this method. By virtue of the generalized Gauss–Radau projection, we can obtain the optimal convergence order in L 2 -norm of O h k + 1 with polynomial of degree k and grid size h. Numerical experiments are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The advection–diffusion–reaction (ADR) systems are often used to describe the chemotaxis models arising in biology. In general, the stiffness from the reaction and diffusion terms often requires very restricted time step size, and the advection term depending on the concentration gradients of another component (the chemoattractant) may lead to sharp peaks in localized spatial regions. It is challenging to design numerical methods that can efficiently handle both difficulties. In this paper, we apply the operator splitting approach to solve the advection–diffusion–reaction systems. In particular, for advection term, we use the positivity-preserving DG method with strong stability preserving (SSP) high order time discretizations. For reaction–diffusion term, direct discontinuous Galerkin (DDG) method is used in spatial discretization and Krylov IIF method is applied in time discretization. Numerical examples are shown to demonstrate the accuracy, efficiency and robustness of the method. [ABSTRACT FROM AUTHOR]

Numerical computations of ground and excited states of Bose–Einstein condensates (BECs) require high spatial resolution due to the presence of the multiscale structures and boundary layers and interior layers in the solution. In this paper, a new discontinuous Galerkin (DG) method is presented for the computation of ground and excited states of BECs. For the spatial discretization, the direct discontinuous Galerkin (DDG) method is applied by using the normalized gradient flow. For the time discretization, we integrate the ordinary differential equations (ODEs) which is obtained by linearizing the interaction term. To evaluate the matrix exponential operator efficiently, we apply the Krylov subspace approximations to the matrix exponential operator. Numerical examples with different potentials are reported to demonstrate the validity and effectiveness of the semi-implicit DG method. [ABSTRACT FROM AUTHOR]

To get the stationary patterns in the approximation of reaction diffusion systems, the computation of evolution equations on large intervals of time is required. In this paper, a new nonlinear Galerkin method based on finite element discretization is presented for the approximation of reaction diffusion equations on large intervals. The new scheme is based on two different finite element spaces defined respectively on one subspace of lower-degree shape function and one of higher-degree shape function. Nonlinearity and time dependence are both treated on the lower-degree space and only a fixed stationary equation needs to be solved on the higher-degree space at each time level. The proof of the stability and convergence of the method is presented in one dimension. Numerical solutions of some well-studied reaction–diffusion systems are presented to demonstrate the effectiveness of the nonlinear Galerkin method. [ABSTRACT FROM AUTHOR]

Abstract: Nonlinear reaction–diffusion systems are often employed in mathematical modeling for pattern formation. Most of the work to date has been concerned within one-dimensional or rectangular domains. However, it is recognised that in most applications multidimensional complex geometrical domains are typically more important. In this paper we solve reaction–diffusion systems by combining direct discontinuous Galerkin (DDG) finite element methods with implicit integration factor (IIF) time integration method, on triangular meshes. This allows us solve the nonlinear algebraic systems on an element-by-element bases with significant gains in computational time. Numerical solutions of two reaction–diffusion systems, the well-studied Schnakenberg model and chloride–iodide–malonic acid (CIMA) reactive model, are presented to demonstrate effects of various domain geometries on the resulting biological patterns. Our numerical results are in good agreement with other numerical and analytical results, and with experimental results. [Copyright &y& Elsevier]

Abstract: In this paper, a numerical solution for the generalized Burger’s–Huxley equation and the generalized Burger’s–Fisher equation are presented by using the local discontinuous Galerkin method. Firstly we rewrite the two kinds of equations into first order systems, then apply the discontinuous Galerkin method to the systems. Secondly, the systems for ordinary differential equations yielded are solved by the third-order Runge–Kutta time integration method. The numerical results obtained by this way are compared with exact solution to demonstrate accuracy and capability of the method. [Copyright &y& Elsevier]

A new combined method is constructed for solving incompressible miscible displacement in porous media. In this procedure, a hybrid mixed element method is constructed for the pressure and velocity equation, while a symmetric discontinuous Galerkin finite element method is proposed for the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also conserves the local mass balance. The stability and consistency of the method are analyzed and the optimal error estimate in L ∞ ( L 2 ) for velocity and concentration and the super convergence in L ∞ ( H 1 ) for pressure are derived. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the system of two-dimensional Burgers’ equations are solved by local discontinuous Galerkin (LDG) finite element method. The new method is based on the two-dimensional Hopf–Cole transformations, which transform the system of two-dimensional Burgers’ equations into a linear heat equation. Then the linear heat equation is solved by the LDG finite element method. The numerical solution of the heat equation is used to derive the numerical solutions of Burgers’ equations directly. Such a LDG method can also be used to find the numerical solution of the two-dimensional Burgers’ equation by rewriting Burgers’ equation as a system of the two-dimensional Burgers’ equations. Three numerical examples are used to demonstrate the efficiency and accuracy of the method. [Copyright &y& Elsevier]