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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]

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]