This paper concerns numerical solutions for a fractional partial differential equation arising as a nonreflecting boundary condition in water wave propagation. The fractional derivative operator is written as divergence of a singular integrable convolution, which allows the equation to be viewed as a conservation law with a linear nonlocal flux. A semi-discrete finite volume scheme is presented, using conservative piecewise polynomial reconstruction of the solution. The convolution with the singular kernel is then integrated exactly. Time integration uses Runge–Kutta schemes of matching order. Stability is discussed, convergence is established and numerical examples are presented.