  In this paper, we provide a systematic methodology for calculating
multi-order asymptotic expansion of blow-up solutions near blow-up for
autonomous ordinary differential equations (ODEs). Under the specific form of
the principal term of blow-up solutions for a class of vector fields, we
extract algebraic objects determining all possible orders in the asymptotic
expansions. Examples for calculating concrete multi-order asymptotic expansions
of blow-up solutions are finally collected.
