Minimal residual multistep methods for large stiff non-autonomous linear problems

The purpose of this work is to introduce a new idea how avoid the factorization large matrices during solution stiff systems ODEs. Starting from general form an explicit linear multistep method we suggest adaptively choose its coefficients on each integration step in order minimize norm residual implicit BDF formula. Thereby reduce number unknowns n O(1), where dimension ODE system. We call type methods Minimal Residual Multistep (MRMS) methods. In case non-autonomous problem, besides evaluations right-hand side ODE, resulting numerical scheme additionally requires one least-squares problem with thin matrix per step. show that and zero-stability properties coincide those used underlying For simplest analog Euler stability are investigated. Though classical absolute analysis not fully relevant MRMS methods, it shown one-step applicable case. experiment section consider fixed-step two-dimensional heat equation using their counterparts. starting values taken preset slowly-varying exact solution. comparison showed both give similar solutions, but faster, advantage considerably increases growth dimension. Python code experimental can be downloaded GitHub repository https://github.com/bfaleichik/mrms.