Sensitivity-based methods for convergence acceleration of iterative algorithms Original Research Article

A new method to accelerate the convergence of iterative schemes for the numerical integration of systems of partial differential equations has been developed. The basic idea is that the residual at a grid point depends on the values of the solution vector at the neighboring grid points used in the local discretization approximation. Thus, the new acceleration method is based on the sensitivity of the future residual to the change in the solution vector at the neighboring grid points with the objective to minimize the future residual. The result is a set of optimum iterative relaxation parameters for the entire flow field or for each individual grid line. The method is easy to implement in the existing codes. We have applied it to a finite difference code for two-dimensional incompressible Navier-Stokes equations. Test cases involve laminar and turbulent flows with severe grid clustering and flow separation. The results are compared with those of a basic explicit Runge-Kutta (RK) time-stepping iterative algorithm and with the implicit residual smoothing (1RS) and the distributed minimal residual (DMR) acceleration techniques. The new acceleration scheme is shown to be superior to these methods especially on highly-clustered grids.