S. Geng, Construction of high order symplectic Runge--Kutta methods, J. Comput. Math. 11 (1993) 250--260. Home/Search Document Not in Database Summary Related Articles Check |
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A Parallelizable Preconditioner for The Iterative Solution of .. - Jay, BRACONNIER (1999) (Correct)
.... the k th shifted Legendre polynomial P k (x) p 2k 1 k Delta d k dx k i x k (x Gamma 1) k j = p 2k 1 k X j=0 ( Gamma1) j k k j j k j x j : For recent references and more details about the W transformation we refer the reader to [20, Section IV.5] and [6, 13]. In the remainder of the article we will assume that X : W T BAW is tridiagonal and D : W T BW is diagonal and nonsingular; two conditions which are satisfied for most IRK methods of interest, such as Gauss, Radau IA IIA, Lobatto IIIA IIIB IIIC IIIC schemes [6, 13, 20, 22] ....
.... IV.5] and [6, 13] In the remainder of the article we will assume that X : W T BAW is tridiagonal and D : W T BW is diagonal and nonsingular; two conditions which are satisfied for most IRK methods of interest, such as Gauss, Radau IA IIA, Lobatto IIIA IIIB IIIC IIIC schemes [6, 13, 20, 22]. For 4 L. O. Jay and T. Braconnier these IRK methods the transformed matrix X and the matrix D read X = 0 B B B B B B B 1=2 Gammai 1 O i 1 0 . Gammai s Gamma2 i s Gamma2 0 fi s Gamma1;s O fi s;s Gamma1 fi ss 1 C C C C C C C A ; D = diag(1; 1; 1; d s ) 6) ....
S. Geng, Construction of high order symplectic Runge-Kutta methods, J. Comput. Math., 11 (1993), pp. 250--260.
Highly Oscillatory Systems and Periodic-Stability - Jay, Petzold (1995) (3 citations) (Correct)
....the matrix e A 0 has s Gamma 2 rows and the matrix f b A 0 has s Gamma 2 columns, we have deg(q( 2s Gamma 4. To show that deg(p 11 ( p 22 ( 2s Gamma 2 we will make use of the W transformation. For details about the W transformation we refer the reader to [15, Section IV.5] and [4, 10]. Here the aim is to express p 11 ( and p 22 ( in terms of the transformed matrices X : W T BAW and b X : W T B b AW where B = diag(b 1 ; b s ) the coefficients of the matrix W are given by w ij = P j Gamma1 (c i ) with P j Gamma1 (x) the (j Gamma 1) th shifted Legendre ....
....W are given by w ij = P j Gamma1 (c i ) with P j Gamma1 (x) the (j Gamma 1) th shifted Legendre polynomial, and the c i are the nodes of the Lobatto quadrature. The matrices X; b X for the Lobatto IIIA and Lobatto IIIB coefficients are given respectively 8 L. O. JAY AND L. R. PETZOLD by [10] X = 0 B 0 X 0 . 0 1 C A = 0 B B B B B B 1=2 Gammai 1 O 0 i 1 0 . 0 . Gammai s Gamma2 . i s Gamma2 0 0 O i s Gamma1 u 0 1 C C C C C C A ; b X = b X 0 0 : 0 = 0 B B B B B B 1=2 Gammai 1 O i 1 0 . Gammai s Gamma2 O i ....
S. Geng, Construction of high order symplectic Runge-Kutta methods, J. Comput. Math., 11 (1993), pp. 250--260.
A Parallelizable Preconditioner for The Iterative Solution of .. - Jay, Braconnier (1999) (Correct)
.... Legendre polynomial P k (x) p 2k 1 k Delta d k dx k i x k (x Gamma 1) k j = p 2k 1 k X j=0 ( Gamma1) j k 0 B k j 1 C A 0 B j k j 1 C Ax j : For recent references and more details about the W transformation we refer the reader to [20, Section IV.5] and [6,13]. In the remainder of the article we will assume that X : W T BAW is tridiagonal; D : W T BW is diagonal and nonsingular; two conditions which are satisfied for most IRK methods of interest, such as Gauss, Radau IA IIA, Lobatto IIIA IIIB IIIC IIIC schemes [6,13,20,22] For these ....
.... Section IV.5] and [6,13] In the remainder of the article we will assume that X : W T BAW is tridiagonal; D : W T BW is diagonal and nonsingular; two conditions which are satisfied for most IRK methods of interest, such as Gauss, Radau IA IIA, Lobatto IIIA IIIB IIIC IIIC schemes [6,13,20,22]. For these IRK methods the transformed matrix X and the matrix D read X = 0 B B B B B B B B B B B B 1=2 Gammai 1 O i 1 0 . Gammai s Gamma2 i s Gamma2 0 fi s Gamma1;s O fi s;s Gamma1 fi ss 1 C C C C C C C C C C C C A ; D = diag(1; 1; 1; d s ) 6) where i k = 1= ....
S. Geng. Construction of high order symplectic Runge-Kutta methods. J. Comput. Math., 11:250--260, 1993.
A parallelizable preconditioner for the iterative solution of .. - Jay, Braconnier (1999) (Correct)
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S. Geng, Construction of high order symplectic Runge--Kutta methods, J. Comput. Math. 11 (1993) 250--260.
Structure Preservation For Constrained Dynamics With Super.. - Jay (1998) (Correct)
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S. Geng, Construction of high order symplectic Runge-Kutta methods, J. Comput. Math., 11 (1993), pp. 250--260.
Highly Oscillatory Systems And Periodic-Stability - Laurent Jay And (1995) (3 citations) (Correct)
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S. Geng, Construction of high order symplectic Runge-Kutta methods, J. Comput. Math., 11 (1993), pp. 250--260.
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