Geodesic
Approximation of the Jacobi matrix in $(m,3)$-methods of solving stiff systems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 11 (2008) no. 3, pp. 283-295.

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The theorem that the maximal order of accuracy of $L$-stable $(m,k)$-methods with freezing the Jacobi matrix is equal to four. 
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A. L. Dvinsky; E. A. Novikov. Approximation of the Jacobi matrix in $(m,3)$-methods of solving stiff systems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 11 (2008) no. 3, pp. 283-295. http://geodesic.mathdoc.fr/item/SJVM_2008_11_3_a3/

[1] Rosenbrock H. H., “Some general implicit processes for the numerical solution of differential equations”, Computer, 5 (1963), 329–330 | DOI | MR | Zbl

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[4] Novikov E. A., Odnoshagovye bezyteratsionnye metody resheniya zhestkikh sistem, Dis. $\dots$ dokt. fiz.-mat. nauk: 01.01.07, Novosibisk, 1991

[5] Novikov E. A., Dvinskii A. L., “$L$-ustoichivaya (5, 2)-skhema chetvertogo poryadka”, Voprosy matem. analiza, Vyp. 8, IPTs KGTU, Krasnoyarsk, 2004, 134–142

[6] Novikov E. A., Golushko M. I., Issledovanie (m, k)-metodov resheniya zhestkikh sistem s tremya vychisleniyami pravoi chasti, Preprint / VTs SO RAN No 5, Krasnoyarsk, 1992

[7] Novikov E. A., Dvinskii A. L., “Zamorazhivanie matritsy Yakobi v (3, 2)-metode resheniya zhestkikh sistem”, Vychislitelnye tekhnologii. — Novosibirsk, 8:3 (2003), 272–278

[8] Dvinskii A. L., Issledovanie (m, k)-metodov s $L$-ustoichivymi promezhutochnymi skhemami dlya resheniya zhestkikh zadach, Diss. $\dots$ kand. fiz.-mat. nauk: 01.01.07, Krasnoyarsk, 2004

[9] Novikov E. A.,Shitov Yu. A., Algoritm integrirovaniya zhestkikh sistem na osnove (m, k)-metoda vtorogo poryadka tochnosti s chislennym vychisleniem matritsy Yakobi, Preprint / VTs SO RAN No 20, Krasnoyarsk, 1988

[10] Golushko M. I., Issledovanie (m, 3)-metodov resheniya zhestkikh sistem, Diss. $\dots$ kand. fiz.-mat. nauk: 01.01.07, Krasnoyarsk, 1993