Efficient ADI and spline ADI methods for the steady‐state Navier‐Stokes equations

The present paper provides an improved alternating direction implicit (ADI) technique as well as high‐order‐accurate spline ADI method for the numerical solution of steady two‐dimensional incompressible viscous flow problems. The vorticity‐stream function Navier‐Stokes equations are considered in a general curvilinear coordinate system, which maps an arbitrary two‐dimensional flow domain in the physical plane into a rectangle in the computational plane. The stream function equation is parabolized in time by means of a relaxation‐like time derivative and the steady state solution is obtained by a time‐marching ADI method requiring to solve only 2 × 2 block‐tridiagonal linear systems. The difference equations are written in incremental form; upwind differences are used for the incremental variables, for stability, whereas central differences approximate the non‐incremental terms, for accuracy, so that, at convergence, the solution is free of numerical viscosity and second‐order accurate. The high‐order‐accurate spline ADI technique proceeds in the same manner; in addition, at the end of each two‐sweep ADI cycle, the solution is corrected by means of a fifth‐order spline interpolating polynomial along each row and column of the computational grid, explicitly. The validity and the efficiency of the present methods are demonstrated by means of three test problems