A method for the treatment of polar singularities of Euler equations is given. The first radial mesh point locates at a half-space away from the centerline. Based on the characteristics of the physical variables,the boundary near centerline is extended so that a high order finite difference scheme can be utilized as at inner mesh points.Similarly,in azimuthal direction the boundary is extended according to the periodicity.An uniform high-order precision is preserved during discretization of equations.

Based on the close relationship between Hamilton-Jacobi (H-J) equations and hyperbolic conservation laws,a high-order numerical method is developed to solve the H-J equations in the 3rd order and 5th order compact schemes.The upwind compact schemes are tested on a variety of one-dimensional and two-dimensional problems,including a problem related to the Richtmyer-Meshkov instability accelerated by planar shocks.Numerical results show that these schemes yield uniform high-order accuracy in smooth regions and satisfactorily resolve discontinuities in the derivatives.Moreover,since the present methods have less numerical dissipation than WENO scheme with the same order,they could be used to solve the H-J equations more accurately in the simulation of multi-scale complex flows.