1. in a thorough, careful, and exact manner
2. in a harsh or severe manner

Examples

• The scientists rigorously tested their hypothesis before publishing results.
• Students must rigorously follow all safety procedures in the laboratory.
• The data was rigorously analyzed using statistical methods.
• The coach rigorously trained the team for the championship.
• All applications will be rigorously reviewed by the committee.
• The building codes are rigorously enforced in this city.