Water wave propagation in unbounded domains. Part II: Numerical methods for fractional PDEs

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.