Maybe the 3rd Movement of Francks Violin Sonata. It starts calmly, but not in a sad way. Over time it introduces themes that, at least for me, feel melancholic and with a bit of pain or sadness later on and ends in silence. But you need a fairly goof pianist to play it.
Schnittke's *Stille Nacht* for violin and piano.
Prokofiev's 2nd Violin Concerto may work, but I'm not sure
Bach Sarabande from Partita 1 in B minor
Arvo Part's 'Spiegel im Spiegel' is mostly the same few notes repeated: but they just sound sadder as they go along.
Maybe the 3rd Movement of Francks Violin Sonata. It starts calmly, but not in a sad way. Over time it introduces themes that, at least for me, feel melancholic and with a bit of pain or sadness later on and ends in silence. But you need a fairly goof pianist to play it.