A High-Order Accuracy Algorithm For Hyperbolic Conserbation Laws

Hyperbolic conservation equation is used widely in transportation, preventing andcontrolling water pollution, weather prediction, resources exploration and so on.However, usually we can't give out the continuous solution in the whole fields. So theresearch for numerical solution is important on the theoretical analysis and engineeringapplication.Finite difference method is an important method to study numerical method. Here,derivatives are approached by difference coefficient, and we will get the analoguesolution of the original problem. Some representative schemes of hyperbolicconservation have one or two order accuracy. In order to improve the accuracy, thedifference schemes need to increase the basepoint number or add additional physicalconditions. In this paper, the high order accuracy scheme, Perturbation FiniteDifference(PFD) is different from the above ideas. PFD Scheme has fewer basepoints,and keep the original boundary treatment in physical terms.Perturbation finite difference format is proposed by GaoZhi researcher of ChineseAcademy of Sciences for the first time. It is a kind of high order accuracy formats.According to the demand of accuracy, we can get it through the truncation error andmodified the original differencl equation. In PFD, derivatives are approached bydifference coefficient. At the same time, expand the coefficients of derivatives or sourceitem in a power series for step length.First, this paper improves the second-order accuracy upwind PFD scheme. Becauseit adopts exact value instead of analogue value, the modified PFD will eliminatederivatives in the denominator. Thus, it expands its applicable scope, and reduce the riskof error diffusion.Second, PFD is being applied to Lax-Friedrichs difference scheme and centraldifference scheme. Lax-Friedrichs difference scheme is first-order accuracy of both timeand space and local stability. Lax-Friedrichs PFD scheme with first-order accuracy oftime and second-accuracy of space is instability, and Lax-Friedrichs PFD scheme withsecond-order accuracy of both time and space has the same stable region ofLax-Friedrichs scheme Central difference scheme with first-order accuracy of time andsecond-accuracy of space is instability, but Central PFD scheme with second-orderaccuracy of both time and space is local stability. This paper gives the stability analysisand points out that stability region is different between the PFD schemes and originaldifference schemes.Finally, this paper constructs a third-order accuracy of time, fourth-order accuracyof space central PFD scheme. Through the examples analysis, we get its stability region.