This report contains lecture notes used for the 2016 edition of the Rome-Moscow school of Matrix Methods and Applied Linear Algebra, held in Moscow and Rome (respectively, in August and September 2016). The notes deal with some matrix analysis problems which arise in construction and analysis of time integration methods for solving large systems of ordinary and partial differential equations (ODEs and PDEs). The material treated includes some aspects of finite-difference approximation of convection–diffusion operators (used, following the framework of the methods of lines, to reduce time-dependent convection–diffusion problems to ODE systems), stability of the ODE systems, the logarithmic matrix norm, stability of the implicit–explicit θ-method, splitting methods, Rosenbrock methods with approximate matrix factorizations and Krylov sub- space exponential time integration.

method of lines, convection–diffusion equation, stability of differential equations and difference schemes, matrix exponent, logarithmic matrix norm, Rosenbrock methods, Krylov subspace

Mathematical problems and theory of numerical methods

Botchev Mikhail Aleksandrovich,  ,  orcid.org/0000-0001-5901-7120,  KIAM RAS