A high accuracy defect-correction multigrid method for the steady incompressible Navier-Stokes equations

The solution of large sets of equations is required when discrete methods are used to solve fluid flow and heat transfer problems. Although the cost of the solution is often a drawback when the number of equations in the set becomes large, higher order numerical methods can be employed in the discretization of differential equations to decrease the number of equations without losing accuracy. For example, using a fourth-order difference scheme instead of a second-order one would reduce the number of equations by approximately half while preserving the same accuracy. In a recent paper, Gupta has developed a fourth-order compact method for the numerical solution of Navier-Stokes equations. In this paper we propose a defect-correction form of the high order approximations using multigrid techniques. We also derive a fourth-order approximation to the boundary conditions to be consistent with the fourth-order discretization of the underlying differential equations. The convergence analysis will be discussed for the parameterized form of a general second-order correction difference scheme which includes a fourth-order scheme as a special case.