Numerical Perturbation Reconstruction Of Third-order WENO Schemes And The Application In Convection Diffusion Equations

Weighted Essentially Non-Oscillatory(WENO) method, which is constructedbased on Essentially Non-Oscillatory(ENO) method, is a kind of high-order accurateand high resolution numerical method. WENO scheme introduces the concept ofweight. In order to achieve high order approximation, it combines all the lowerorder approximations on candidate stencils by multiplying their weights respectively.The size of weight on each stencil is determined by their smoothness. The weightson discontinuous stencils are approximately equal to zero. However, in smoothregions, the nonlinear weights which are computed by smoothness indicators areapproximately equal to ideal weights. Furthermore, in discontinuous regions, WENOscheme maintains the property of stable, high resolution and non-oscillatory.Numerical perturbation algorithm is a new method to construct high-orderaccurate CFD scheme. While maintaining the original discrete scheme unchanged,the non-derivative terms are perturbationally reconstructed by the power series ofcell size. Then, we deduce to obtain its modified diferential equation and calculateperturbation coefcients by eliminating its truncation error. The perturbationaloperation raises the original scheme’s accurate and reduces numerical dissipation.Convection-difusion equations are fundamental equations. We usually usethese kinds of equations to describe heat exchange and the distribution of airpollutants and river pollutants. It play an important role in viscous fluid dynamicsand other applications. In this article we reconstruct WENO scheme by Numericalperturbation algorithm to solve convection-difusion equations. In this paper wemainly do the following two works:The first work is: we perturbationally reconstructed the discretization of three-order finite diference WENO scheme based on convection-difusion equation withoutsource term. Global perturbation (using information of all nodes) and upstream anddownstream perturbation (using the law of that the downstream does not afect theupstream)are both reconstructed. In this chapter We use two numerical tests toexamine errors and orders of the new perturbation schemes. The numerical resultsshow that the errors are obviously reduced and the orders are raised.The second work is: we use three-order WENO scheme and2-CDS to discretizeconvection-term and difusion-term respectively. Like the first work, we can deduceto obtain global and upstream and downstream perturbation scheme. Because of the source terms in convection-difusion equation, the linear relationship betweenhigh-order derivative and first-order derivative tends to be more complex. Thenit makes the process of perturbation reconstruction more difcult. At the end ofchapter three, two numerical tests are used to examine the errors and orders of thenew perturbation schemes. The numerical results show that our reconstruction isefective.