Random Perturbations On The Convection-diffusion Equation And Its Applications

It may cause the instability of the numerical mode, even may cause the problem the numerical solutions were completely distorted, because of the stochastic disturbance or uncertainty exist in borders or parameters. Therefore, by solving convection-diffusion equation, this paper researched the influences on the numerical solution of several difference schemes under the conditions of the stochastic disturbed boundaries and parameters.The results manifest, the four schemes, which are central explicit scheme, modified central explicit scheme, exponent scheme, upwind scheme, respectively, all could be applied into numerically solving the convection-diffusion equation satisfactory. However, the worst results, which might most probably cause the instability of the mode, were obtained by the exponent scheme because of the stochastic disturbed boundaries and parameters. Generally, we should seldom use the exponent scheme to disperse the convection-diffusion equation. Moreover, with the increase of the number of grid point, the stabilities of calculation of the four schemes have got very great improvement. As a result, the influences for the stochastic disturbed boundaries and parameters might be suppressed or even eliminated by increasing the number of the grid point in numerical calculation.A quasi-wavelet numerical method (QWNM) is introduced for solving the convection-diffusion equation, and compared its computational results with the up-wind scheme. It is found that the QWNM is able to numerically solve the convection-diffusion equation satisfactory. The accuracy of the convection-diffusion equation by using the QWNM is relatively high, and better than that of the up-wind scheme. The results manifest, the calculating bandwidth W has an extremum, when the bandwidth W fetches the value, the precision of the quasi-wavelet solution of theconvection-diffusion equation is relatively high, and better than the up-wind scheme.When the parameter and initial conditions are stochastically disturbed, the precision of the quasi-wavelet solution of the equation is smaller than that of the up-wind scheme. Under the condition of stochastic boundary disturbances, the results of the quasi-wavelet solution are the same with that of the up-wind scheme when the calculating bandwidth W fetches properly value. Therefore, it is necessary in the practical application of the QWNM to properly select the computational bandwidth according to parameter, boundary and initial conditions so as to raise the accuracy of the numerical solution as well as to reduce the influence of stochastic disturbances on the numerical solution.In view of the stochastic errors exist in the observation, which have great influence on Numerical Weather Forecast, we make use of the Monte-Carlo simulation method to gain a series of initial values sets, about 2000 initial values in each set. By investigating the evolution of the numbers of the particles, which falls into the dynamic window we defined, and the average forecast's weight X of the Lorenz system, we researched preliminarily the predictability of the Lorenz system. The results manifest, the evolution of the numbers reflects the predictability of the Lorenz system to some certain extent, and different predictability of the initial values sets in different areas. Furthermore, we found that there are different characters between three different unstable equalization's points.