Remove stems and pit the sour cherries over a bowl to catch their juices.
In a medium bowl, combine the pitted cherries, granulated sugar, and lemon juice. Toss to coat and let macerate until the sugar dissolves and juices pool, 15–20 minutes.
Scrape the cherry mixture and accumulated juices into a medium saucepan and add the water.
Bring to a gentle simmer over medium heat, then reduce to medium-low and cook, stirring occasionally and skimming any foam, until the cherries are tender but intact and the syrup lightly coats a spoon, 8–12 minutes.
Remove from heat and let cool 10–15 minutes; the compote will thicken slightly as it cools. Taste and adjust with a little more sugar or lemon juice if needed.
Serve warm or chilled. Transfer to a clean jar, cover, and refrigerate up to 1 week.