SOR Iteration Research Articles

An Analysis of the Method of Lines for the Reynolds Equation in Hydrodynamic Lubrication

The Reynolds equation for a hydrodynamic journal bearing is discretized with the method of lines. A continuous analogue of the line SOR iteration is set up to solve the resulting free multipoint system of ordinary differential equations via a sequence of one-dimensional free boundary problems. It is shown that each one-dimensional problem can be solved with invariant imbedding, that the line SOR method converges, and that the solution of the by-lines discretization converges to the solution of a variational inequality derived from the Reynolds equation.

The method of lines for Poisson's equation with nonlinear or free boundary conditions

The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.

Round-off error analysis of iterations for large linear systems

We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A *>0 andA has PropertyA. This means that the computed resultx k approximates the exact solution ? with relative error of order ? ?A?·?A ?1? where ? is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector ?Ax k ?b? is of order ? ?A?2 ?A ?1? ??? and hence the remaining three iterations arenot well-behaved.

Numerical solutions for viscous and potential flow about arbitrary two-dimensional bodies using body-fitted coordinate systems

A procedure for numerical solution of the time-dependent, incompressible Navier-Stokes equations for the flow about arbitrarily shaped two-dimensional bodies is given. This solution is based on a technique of automatic numerical generation of a curvilinear coordinate system having a coordinate line coincident with the body contour regardless of its shape. The implicit solution utilizes the vorticity-stream function formulation with a false-position iterative adjustment of the surface vorticity in satisfaction of the no-slip boundary condition. All field equations are approximated using central differences and are solved simultaneously at each time step by SOR iteration. Excellent agreement with the Blasius boundary layer solution is obtained for a semi-infinite flat plate. Results are presented for Reynolds numbers up to 2000 for several airfoils and a cambered rock. A potential flow solution based on the same coordinate systems is also given, and excellent agreement with analytic solutions for Karman-Trefftz airfoils is shown.

Iterative eigenvalue algorithms based on convergent splittings

Application of direct iterations, based on convergent splittings, to the eigenvalue problem of large sparse symmetric matrices is discussed. A general convergence proof is given, and it is shown how parameters should be chosen to give the best rate of convergence. As special examples are considered, SOR iteration and iteration based on the use of a fast direct Poisson solver. Numerical tests are reported.

The Determination of Consistent Orderings for the SOR Iterative Method

The Determination of Consistent Orderings for the SOR Iterative Method

On the Convergence of SOR Iterations for Finite Element Approximations to Elliptic Boundary Value Problems

SOR iterations for linear systems arising from finite element approximations to $2m$th order elliptic problems are considered. It is shown that, in general, the asymptotic rate of convergence is $O(h^{2m} )$, where h denotes the mesh length. However, if the relaxation parameters are suitably chosen, this can be increased to $O(h^m )$.