A new discretization method and its application to solve incompressible Navier-Stokes equations

A new numerical method is presented in this paper. This method directly sol ves partial differential equations in the Cartesian coordinate system. It c an be easily applied to solve irregular domain problems without introducing the coordinate transformation technique. The concept of the present method is different from the conventional discretization methods. Unlike the conv entional numerical methods where the discrete form of the differential equa tion only involves mesh points inside the solution domain, the new discreti zation method reduces the differential equation into a discrete form which may involve some points outside the solution domain. The functional values at these points are computed by the approximate form of the solution along a vertical or horizontal line. This process is called extrapolation. The fo rm of the solution along a line can be approximated by Lagrange interpolate d polynomial using all the points on the line or by low order polynomial us ing 3 local points. In this paper, the proposed new discretization method i s first validated by its application to solve sample linear and nonlinear d ifferential equations. It is demonstrated that the present method can easil y treat different solution domains without any additional programming work. Then the method is applied to simulate incompressible flows in a smooth ex pansion channel by solving Navier-Stokes equations. The numerical results o btained by the new discretization method agree very well with available dat a in the literature. All the numerical examples showed that the present met hod is very efficient, which is suitable for solving irregular domain probl ems.