Compact WENO Methods For Nonlinear High-order Equations

Weighted essentially non-oscillatory(WENO)methods are one of the most effective numerical methods in computational fluid dynamics.The main idea of WENO methods is to obtain a high-order approximation through a convex combination of low-order numerical fluxes,and to solve discontinuities in an essentially non-oscillation fashion.Since WENO schemes are high-order accurate and high resolution,they are widely used in shock wave turbulence,aeroacoustics and other fields.However,the traditional WENO schemes need a large stencil,which makes them difficult to extent high-dimensional problems and to treat complicated boundaries.Therefore,compact high-order nonlinear schemes have attracted widespread attention in recent years.This paper is mainly to design compact WENO methods for nonlinear high-order equations,including:(1)For nonlinear degenerate parabolic equations,we design a com-pact WENO finite difference method which directly approximates the second derivative.Through theoretical analysis and classical numerical experiments,we have proved the new scheme obtains sixth-order accuracy in smooth areas and non-oscillation near discontinu-ities.The linear weights associated can be any set of positive real number,add up to 1.(2)For the Degasperis-Procesi equation,which is an approximate model of shallow water wave propagation in the small amplitude and long wavelength regime,we rewrite it into a hyperbolic-parabolic system.A compact WENO method that approximates the first deriva-tive is adopted.This scheme uses a five-point stencil,reconstructed from a fourth-degree polynomial and two first-degree polynomials.Through classical numerical experiments,we have proved that this method can simulate discontinuous shock wave solutions and peak solutions well,and can obtain fifth-order accuracy in smooth areas.Moreover,we have more freedom to select linear weights using this compact method,compared with the traditional one.