Geodesic
On analytical families of matrices generating bounded semigroups
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 3-16 
P. A. Bakhvalov; M. D. Surnachev. On analytical families of matrices generating bounded semigroups. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 1, pp. 3-16. http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a1/
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     author = {P. A. Bakhvalov and M. D. Surnachev},
     title = {On analytical families of matrices generating bounded semigroups},
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     url = {http://geodesic.mathdoc.fr/item/SJVM_2021_24_1_a1/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider linear schemes with several degrees of freedom (DOFs) for the transport equation with a constant coefficient. The Fourier transform decomposes the scheme into a number of finite systems of ODEs, the number of equations in each system being equal to the number of DOFs. The matrix of these systems is an analytical function of the wave vector. Generally such a matrix is not diagonalizable and, if it is, the diagonal form can be non-smooth. We show that in a 1D case for $L_2$-stable schemes the matrix can be locally transformed to a block-diagonal form preserving the analytical dependence on the wave number.

[1] T. Kato, “Perturbation theory for linear operators”, Grund. math. Wiss., 132, Springer, 1966, 477–513 | MR

[2] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[3] H. Baumgärtel, Analytic perturbation theory for matrices and operators, Birkhäuser Verlag, Basel–Boston–Stuttgart, 1985 | MR | Zbl

[4] L. R. Volevich, A. R. Shirikyan, “Remarks on Strongly Hyperbolic Matrices”, Keldysh Institute preprints, 1999, 009

[5] F. Delebecque, “A reduction process for perturbed Markov chains”, SIAM J. Appl. Math., 43:2 (1983), 325–350 | DOI | MR | Zbl

[6] H. O. Kreiss, “Über Matrizen die beshränkte Halbgruppen erzeugen”, Mathematica Scandinavica, 7 (1959), 71–80 | DOI | MR | Zbl

[7] A. H. Levis, “Some computational aspects of the matrix exponential”, IEEE Trans. Automatic Control, 14:4 (1969), 410–411 | DOI | MR

[8] K. Kurdyka, L. Paunescu, “Hyperbolic polynomials and multiparameter real analytic perturbation theory”, Duke Mathematical J., 141:1 (2008), 123–149 | DOI | MR | Zbl

[9] O. N. Kirillov, “Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices”, Zeitschrift für angewandte Mathematik und Physik, 61:2 (2010), 221–234 | DOI | MR | Zbl