In search of optimal acceleration approach to iterative solution methods of simultaneous algebraic equations

This paper presents several new proposals for acceleration of iterative solution methods of both linear and non-linear Simultaneous Algebraic Equations (SAE). The main concept is based on the successive over-relaxation technique (SOR). A new simple and effective way of evaluation of the relaxation parameter is based on either minimization or annihilation of the subsequent solution residuum. The other concept effectively uses features of the infinite geometrical progression. Its quotient is built using solution increments in several initial series of subsequent iterative steps. Both acceleration mechanisms were also combined in order to obtain the best acceleration of the solution process for Simultaneous Linear Algebraic Equations (SLAE). These concepts were tested on many 1D and 2D benchmark problems, with banded and/or sparse systems. For the relaxed Gauss–Seidel (G–S) approach, the convergence rates were up to 200 times better when compared with the standard G–S one. Significant convergence improvement was also reached while testing non-linear SAEs (with the relaxed Newton–Raphson method). The numerical models of the selected engineering problems were based on the meshless approach, due to their more sophisticated nature (when compared e.g. with the finite element analysis).