A high-order scheme for solving wave propagation problems via the direct   construction of an approximate time-evolution operator
  The manuscript presents a new technique for computing the exponential of skew-Hermitian operators. Principal advantages of the proposed method include: stability even for large time-steps, the possibility to parallelize in time over many characteristic wavelengths, and large speed-ups over existing methods in situations where simulation over long times are required. Numerical examples involving the 2D rotating shallow water equations and the 2D wave equation in an inhomogenous medium are presented, and the method is compared to the 4th order Runge-Kutta (RK4) method and to the use of Chebyshev polynomials. Is is demonstrated that the new method achieves high accuracy over long time intervals, and with speeds that are orders of magnitude faster than both RK4 and the use of Chebyshev polynomials. 
T. S. Haut, T. Babb, P. G. Martinsson, B. A. Wingate
math.NA