Godunov's most influential work is in the area of applied and numerical mathematics, particularly in the development of methodologies used in Computational Fluid Dynamics (CFD) and other computational fields.

In this method, the conservative variables are considered as piecewise constant over the mesh cells at each time step and the time evolution is determined by the exact solution of the Riemann (shock tube) problem at the inter-cell boundaries (Hirsch, 1990).

On 1–2 May 1997 a symposium entitled: Godunov-type numerical methods, was held at the University of Michigan to honour Godunov.

These methods are widely used to compute continuum processes dominated by wave propagation.

On the following day, 3 May, Godunov received an honorary degree from the University of Michigan.