[1] Gustafsson is known for his work in numerical methods for time-dependent partial differential equations and its applications in fluid dynamics.

[2][3] Gustafsson has also authored a couple of books on the topic numerical methods applied to PDE.

In 1962, he joined Uppsala University, where there was a tradition of prominent mathematicians, including Arne Beurling and Lennart Carleson.

Gustafsson has continued to pursue numerical methods in differential equations for most of his career, and much of his early work consisted of applying his theoretical developments to practical problems.

These included weather prediction, both in Sweden and at the National Center for Atmospheric Research based in Boulder, Colorado.