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Numerical Analysis and Applications

2021 year, number 1

On analytical families of matrices generating bounded semigroups

P.A. Bakhvalov, M.D. Surnachev
Keldysh Institute of Applied Mathematics, Moscow, Russia
Keywords: spectral analysis, difference scheme, Riesz projection, matrix transform, block diagonalization


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 OFs. 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 L2-stable schemes the matrix can be locally transformed to a block-diagonal form preserving the analytical dependence on the wave number.