Vout = Piecewise[{{1.1182586384 t, 0 <= t <= 1}, {1.1182586384,
    1 <= t <=
     14}, {644.4164053518319` (0.0017353044229987855` -
       0.0017353044229987855` (-14 + t)), 14 <= t <= 15}}]
NDSolve[{D[P[x, t], t] + 7.173581429624171`*^8 D[V[x, t], x] == 0,
  D[V[x, t], t] + D[P[x, t], t]/1000 ==
   Piecewise[{{-0.00984251968503937` Abs[V[x, t]] V[x, t],
      0 < x < 3100}, {-0.006561679790026247` Abs[V[x, t]] V[x, t],
      3100 < x < 5800}, {-0.022497187851518562` Abs[V[x, t]] V[x, t],
      5800 < x < 5809.144}}],
  V[x, 0] == 0.`, P[x, 0] == 3.4474`*^7, P[0, t] == 3.4474`*^7,
  V[5809.144`, t] == Vout}, {P, V}, {x, 0, 5809.144`}, {t, 0, 15}]