How the time a tithi ends is found
The end time of a tithi is the moment when the difference between the Moon's and the Sun's longitudes reaches the next multiple of 12 degrees. The site finds this moment by searching the model's values.
What it is
Because the Moon's speed varies, the end time is not a fixed interval. The method evaluates the elongation at a time, then narrows the search by iteration until the boundary is found to the second.
How it is worked out or read
The start of the next tithi is the same moment, so the end time of one is the start of the next. The panchang shows the end time for the tithi active at sunrise.
Where it appears on this site
The end time is shown with the tithi in the panchang. See tithi explained for the definition.
What has to be found
A tithi is a stretch of twelve degrees in the angle between the Moon and the Sun, measured from the Sun eastward to the Moon. The first tithi runs from 0 to 12 degrees, the second from 12 to 24, and so on up to the thirtieth, from 348 to 360. So the end of a tithi is the moment when that angle reaches the next multiple of twelve degrees. Finding it means finding when a smoothly changing quantity crosses a given value.
The method used here
This site finds the end time in two stages. First it computes the Moon-Sun angle at sunrise and then every thirty minutes after, until the angle passes the next boundary. Then it narrows down the crossing by bisection: it checks the midpoint of the thirty-minute interval, keeps whichever half contains the crossing, and repeats until the interval is one minute long. The same method is used for the nakshatra, the yoga and the karana, each with its own boundary rule. The Method page describes this in full.
Why the ayanamsa does not matter here
The tithi depends only on the difference between the Moon's and the Sun's longitudes. Subtracting the same ayanamsa from both leaves the difference unchanged, so tithi end times are identical whichever ayanamsa is chosen, and identical in tropical and sidereal systems. This is a useful check: if two panchangs disagree about a tithi end time by more than a minute or two, the cause is not the ayanamsa. The nakshatra, by contrast, depends on the Moon's sidereal longitude alone, so its end time does change with the ayanamsa.
One moment, many clocks
A tithi ends at the same instant everywhere on Earth. What differs between places is the local clock time of that instant and its relation to local sunrise. A tithi ending at 6:10 IST ends after sunrise in Kolkata but before sunrise in Mumbai on many days of the year, so the two cities can have different tithis at sunrise. This is why the same festival can fall on different civil dates in different places.
Skipped and repeated tithis
Because tithis vary in length from about 19 to about 26 hours, a short tithi can begin after one sunrise and end before the next, so it is never current at sunrise and is skipped in the day count, called kshaya. A long tithi can span two sunrises and so is repeated, called vriddhi. The scan from sunrise in the method above shows these cases clearly: a skipped tithi appears as two changes within one day.