A Class Of High-precision Numerical Difference Format

In this paper,high order accurate difference schemes following the general explicit difference formulas are developed.Based on the Taylor series expansion,the formulas is obtained by the character of the Vandermonde determinant. Because grid points can construct species order schemes,the different sorts of the same order scheme are compared by the linear advection equation with the two different initial conditions and the upwind-biased schemes with the best accuracy are obtained. A standard Von Neumann Stability analysis is performed on the ten schemes.The results suggest that an even-order scheme decreases the amplitude error of the next lower-order odd scheme,but phase errors are slightly phase accuracy but increase amplitude errors over the next lower-order even scheme.The dispersion and dissipation errors of upwind-biased finite difference schemes are assessed and compared by means of a Fourier analysis of the difference schemes. there is an increase of the dissipation error in the high-wave range compared to the corresponding upwind-biased scheme.However,the increase of numerical dissipation in the low-wave number range is very small. Considering the shallow water equations is becausing the space difference approximations generate the phase errors.Numerical results show that the sixth order upwind-biased scheme is the best agreement with the exact solution. Numerical results will be presented to examined the performance of the newly proposed upwind-biased schemes.first,consider one-dimensional advection in a constant velocity field.The threee types of test problems that are used to evaluate the accuracy of a numerical scheme are the Gaussian function,the square wave function,the triangular function.Each of these functions helps to illustrate some strengths and limitations of a numerical method.We present results of several two-dimesional rotational flow field tests.As initial conditions we use three different test functions:the cone,the cube,and the grooved cylinder.Numerical results show that the six order upwind-biased schems provides highly accurate solutions both in regions where the transported flow variable is smooth and in the vicinity of sharp gradients.In strong deformational flow test,upwind-biased schemes generate the stability numerical results for short integration times. The first successful numerical weather prediction experiments were made using a model based on the barotropic vorticity equation by Charney.24 and 48 hours ahead forecasts of 500hPa geopotential height by the upwind-biasd scheme.Numerical results show that 500hPa geopotential height by the sixth order upwind-biased scheme agree with the real results. Finally,the performance of the symmetric schemes are compared.For the linear advection equation,the symmetric schemes generate the obvious dispersive errors. There is a drastic improvement when going from second to further order,while further changes are moderate.The symmetric schemes are applied to the shallow water equations.Numerical results shows that the symmetric schemes preserve the perfect conservation of total energy,potential enstrophy and mass.