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In this paper, we study the ground and first excited states of the fractional Bose–Einstein condensates (BEC) which is modeled by fractional Gross–Pitaevskii (GP) equation. We first introduce the normalized gradient flow method and prove its energy diminishing property. Then the weighted shifted Grünwald–Letnikov difference (WSGD) method is used to discretize the Gross–Pitaevskii equation. The corresponding normalization and energy diminishing property for the semi-discrete scheme are proved. For the time discretization, we use the implicit integration factor (IIF) method which decouples the diffusion and nonlinear terms separately. Finally the numerical methods are applied to compute the ground and first excited states of fractional BEC with harmonic oscillator, harmonic-plus-optical lattice and box potential. Our numerical results show that the ground and excited states in fractional GP equation differ from those of the standard (non-fractional) GP equation. [ABSTRACT FROM AUTHOR]

In this study, we present a conservative Fourier spectral collocation (FSC) method to solve the two-dimensional nonlinear Schrödinger (NLS) equation. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. Using the spectral differentiation matrices, the NLS equation is reduced to a system of nonlinear ordinary differential equations (ODEs). The compact implicit integration factor (cIIF) method is later developed for the nonlinear ODEs. In this approach, the storage and CPU cost are significantly reduced such that the use of cIIF method becomes attractive for two-dimensional NLS equation. Numerical results are presented to demonstrate the conservation, accuracy, and efficiency 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]

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]

Chemotaxis models are described by a system of coupled nonlinear partial differential equations: a convection–diffusion type cell density equation and a parabolic or reaction–diffusion type chemical concentration equation. This paper is focused on development of a new numerical method for these models. In this method, a splitting mixed element technique is used to solve the chemical concentration equation, where the LBB condition is not necessary and the flux equation is separated from the chemical concentration equation. Moreover a mass-conservative characteristic finite element method is used to solve the cell density equation, avoiding nonphysically oscillations and keeping mass balance globally. The coefficient matrices of the discrete system are symmetric and positive definite. The convergence of this method is analyzed and the corresponding error estimates are derived for pre-blow-up time since we assume boundedness of the exact solution. Numerical simulations are performed to verify that our method can obtain accurate, nonnegative, oscillation-free solutions and has the ability to recover blowing-up solutions. [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]