There's an important semantic distinction here. I think the OP understands this, but it should be highlighted for references: This passage isn't just about arpeggiation; it's about a specific cross-string bowing technique. For instance, Paganini's fifth caprice for solo violin starts with some big A-minor arpeggios; as Flesch's fingerings here show, there's no doubt that fingering changes during the arpeggio:
This passage from the end of the Mendelssohn cadenza, though, are an example of a technique that I don't have a good name for. I've seen it referred to as bariolage and ondeggiando, but technically "bariolage" applies to passages that use different strings primarily to highlight their differences in timber, and "ondeggiando" can apply to two-string patterns.
Whatever you call it, the point is that you voice chords in ways that put a note on each of the four strings, and then rapidly roll the bow across all four strings. The effect can be very arresting, since one can roll the bow much more quickly than one can change left-hand fingers, so you can spray out scintillating showers of 16th notes while only moving your left hand perhaps once per beat or per bar. Another prominent example is in Bach's "Chaconne" from the D minor solo violin partita. In this case, Bach starts out by notating all the pitches, then starts just writing stacked chords with the indication "arpeggio," assuming that you can see from the example how to break them apart.
This passage presents challenges to interpretation. For one thing, he starts with a "zig-zag" pattern (arguably "double ondeggiando" juxtaposing two overlapping pairs of strings) rather than arpeggiating straight up and then down, but later chords don't accommodate that (and are traditionally done in the same style as the Mendelssohn example), and some chords have three members and some four (do you play triplets? 16ths? Do you double a chord member, or "hop" the bow over a non-adjacent string?).
For a composer, you could take the same approach and leave fingering and voicing up to the performer, or you could spell it out. In any case Peter's comment is important: get the input of an instrumentalist. This cross-string technique is highly idiomatic to bowed string instruments, and while we can "cheat" in a number of ways (yes, you can change fingering within the chord, but you don't want to; yes, you can put two notes on one string or skip a string, but it seriously challenges the bow articulation), you are choosing a technique in which the physicality of the instrument dictates the voicing, and might as well write idiomatically. Mendelssohn worked closely with the violinist for whom he was writing the concerto, Ferdinand David. The fact that Mendelssohn notated a cadenza at all was a bit of a break with precedence. A Mozart concerto would expect the performer to improvise a cadenza. I'm having trouble finding the citation for this, but if I recall, Mendelssohn was reluctant to provide a cadenza himself and David encouraged him to, but the result was the product of a lot of correspondence between both, and just as much David's work as Mendelssohn's. (This autograph score shows an early version; it does contain the arpeggiation, though like Bach he notates it as stacked chords, but David eventually stretched it to a much longer cadenza.
