Interpolated differential operator (IDO) scheme for solving partial differential equations

We present a numerical scheme applicable to a wide variety of partial differential equations (PDEs) in space and time. The scheme is based on a high accurate interpolation of the profile for the independent variables over a local area and repetitive differential operations regarding PDEs as differential operators. We demonstrate that the scheme is uniformly applicable to hyperbolic, ellipsoidal and parabolic equations. The equations are solved in terms of the primitive independent variables, so that the scheme has flexibility for various types of equations including source terms. We find out that the conservation holds accurate when a Hermite interpolation is used. For compressible fluid problems, the shock interface is found to be sharply described by adding an artificial viscosity term.