The phrase "vigorous simmer" means to cook something over medium to high heat with a steady, bubbling motion in the liquid, but not boiling too vigorously. Full definition
Reduce heat to maintain a vigorous simmer. (foodandwine.com)
Add the broth, water, reserved mushroom soaking liquid, mirin, soy sauce, vinegar, and chile paste, and bring to a vigorous simmer over high heat. (finecooking.com)
Bring to a boil, then reduce heat to a vigorous simmer, skimming off any white foam that rises to the top. (holajalapeno.com)