A Hybrided Trapezoidal-Difference Scheme for Nonlinear Time-Fractional Fourth-Order Advection-Dispersion Equation Based on Chebyshev Spectral Collocation Method

In this paper,we firstly present a novel simple method based on a Picard integral type formulation for the nonlinear multi-dimensional variable coefficient fourthorder advection-dispersion equation with the time fractional derivative order a2(1,2).A new unknown function v(x,t)=■u(x,t)/■t is introduced and u(x,t)is recovered using the trapezoidal formula.As a result of the variable v(x,t)are introduced in each time step,the constraints of traditional plans considering the non-integer time situation of u(x,t)is no longer considered.The stability and solvability are proved with detailed proofs and the precise describe of error estimates is derived.Further,Chebyshev spectral collocation method supports accurate and efficient variable coefficient model with variable coefficients.Several numerical results are obtained and analyzed in multi-dimensional spatial domains and numerical convergence order are consistent with the theoretical value 3-a order for different a under infinite norm.

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach,the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method(CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.

Convergence analysis for delay Volterra integral equation

In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1, 1]. Then we use the Gauss quadrature formula to approximate the solution. With the help of lemmas, we get the result that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norm. Some numerical experiments are given to confirm our theoretical prediction.