By Erich Bohl (auth.), Rainer Ansorge, Willi Törnig (eds.)
Read or Download Numerical Treatment of Differential Equations in Applications: Proceedings, Oberwolfach, Germany, December 1977 PDF
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Extra info for Numerical Treatment of Differential Equations in Applications: Proceedings, Oberwolfach, Germany, December 1977
Example text
In this case w e use d i f f e r e n t r e l a x a t i o n factors for the s u b m a t r i c e s A 1~ A 2 and A3~ where the o p t i m u m r e l a x a t i o n factor &l best for the m a t r i x type A 1 is d e t e r m i n e d by Y o u n g . We are not so much concerned with i s o l a t e d c a s e s j where the SOR m e t h o d d o e s not converge at a l l . Instead, our p r o b l e m l i e s in the large number of i t e r a t i o n s and therefore in the large c o m p u t a t i o n t i m e required for a g i v e n error r e d u c t i o n , as there is no m e t h o d to d e t e r m i n e the o p t i m u m r e l a x a t i o n factor for the m a t r i c e s of type A 2 or type A 3 .
Ji 2 - n o r m (2') with theorem. - better , a unique exists with solution and we the c o n s t a n t u E C 2Lmax+2 ilU Thus we 3 approximation Let and h e n c e stability finite difference relative to the i/c 2 . Lmax = m a x { L x , L t} Vii 2 ~< K ( A t 2 L t + A x 2Lx) stability V of the obtain can be Then . achieved if the boundary values are known. 3 Now we assume that u(x,t) : -u(x+ Further, i . e. i 0 < x ~< ~ . 3 Nomenclature for the five-point formula (in two dimensions) 39 1] ]]I % A1 ~K,1 SN Hx,1 ! n HX,CI Hy,1 Hy, p A2 Z,1 H_~y_ H x,%l. Hx,Nx Hy, x e 111 A3 I"iy0Ny Hz,v _Hz__,N_z................. Figure /. Schematic representation of the coefficient matrix 40 Literature /1/ Sommerfeld, A. M e c h a n i k der d e f o r m i e r b a r e n Medien AVG, L e i p z i g 1945. /2/ MOiler, W . / W o l f f , W. Numer ische Berechnung dreidi m e n s i o n a l e r M a g n e t felder for groSe Turbogeneratoren bei feldabh~ngiger P e r m e a b i l i t A t und b e l i e b i g e r S t r o m v e r t e i l u n g ETZ A (1973) 5, S° 2 7 6 - 2 8 2 .