Optimum Solution Techniques for Finite-Difference Equations

The results of investigation into the best digital computer solution method for finite-difference approximations to plane stress problems are presented. A general discussion of the problem and the criteria for choosing solution techniques for large systems of simultaneous equations is followed by the introduction of a new space-saving method for representing finite-difference equations. Called diagonal subscripting, it allows the coefficients of a set of 144 finite-difference equations to be stored in approximately 2,000 words of computer memory. Four methods (conjugate gradients, ordinary Gauss-Seidel iteration, accelerated Gauss-Seidel iteration, and Gaussian elimination) are described and compared from the standpoint of accuracy, computer time and storage ; requirements, and ease of programming. It is concluded that elimination is the best of the four methods for solving sparse systems of finite-difference equations, and that a slight modification of the elimination scheme provides an extremely efficient method for evaluating determinants of finite-difference matrices.