The iterative solution of non-linear systems of equations arising from systems of hyperbolic, time-independent partial differential equations (PDEs) is studied. As an important application, the non-linear Euler equations describing inviscid fluid flow in a channel are solved. The PDE is discretized using a finite difference approximation on a structured grid, and the iteration to steady-state is performed using standard explicit Runge-Kutta methods.
A convergence acceleration technique where a semicirculant (SC) approximation of the spatial difference operator or Jacobian is employed as preconditioner is considered. Numerical experiments show that, for advancing the solution in pseudo-time, the single-stage Runge-Kutta method (the forward Euler scheme) is the most efficient. Furthermore, the step in pseudo-time can be chosen as a constant, independent of the number of grid points and the artificial viscosity parameter. These results are also consistent with analytical results for a linear, scalar model problem.
The results for the SC method is compared to those of a standard multigrid (MG) scheme. The number of iterations and the arithmetic complexities are considered, and it is clear that the SC method is more efficient for the problems studied. Also, the performance for the MG scheme is sensitive to the amount of artificial dissipation added to the difference approximation, while the SC method is not.