Numerical Solution of Partial Differential Equations: by L. Collatz (auth.), J. G. Gram (eds.)

By L. Collatz (auth.), J. G. Gram (eds.)

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Extra info for Numerical Solution of Partial Differential Equations: Proceedings of the NATO Advanced Study Institute held at Kjeller, Norway, August 20–24, 1973

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Hageman, L. , and R. B. Kellogg, "Estimating optimum overrelaxation parameters," Math. Compo 22, 60-68 (1968). 7. Kahan, W. "Gauss-Seidel methods of solving large systems of linear equations," doctoral thesis, University of Toronto, Toronto, Canada (1958). 8. Kulsrud, H. E. "A practical technique for the determination of the optimum relaxation factor of the successive overrelaxation method," Comm. Assoc. Comput. Mach. 4, 184-187 (1961). 9. Reid, J. K. "A method for finding the optimum successive over-relaxation parameter," The Computer Journal 9, 200-204 (1966).

15). 4b. Semi-iteration (alternative to 4a). 2) with a = 0, ~ = S(~l)' 5. Convergence tests. 6) lliill ~ < A If this test is satisfied t· The semi-iterative method is carried out until < 1:. 8 ) 9. -ull A~ NUMERICAL RESULTS We now describe some numerical experiments which were carried out using the accelerated SSOR method. 2) was solved in the unit square with boundary values zero on SOLUTION OF LINEAR SYSTEMS OF EQUATIONS 51 all sides except where y = O. On the side y = 0, the values of u(x,y) were unity.

3) are the velocity components. h and ~ are the total energy and entropy per unit mass respectively. 3) with P the denSity and T the temperature. The function :; p = P(Qi,h,~) = ~ Qi vi + p i=l is introduced. where ~ = vi (i = 1,2,3), : = p, ~ =- pT. The dual variational principles involving p and P respectively can be strengthened to extremum principles for particular types of compressible flow. VARIATIONAL PRINCIPLES - A SURVEY 31 Two recent accounts attempting to unify dual principles are due to Sewell [lOJ and Arthurs [lJ.

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