Higher order finite difference methods for wave propagation problems

Wave propagation is described by the wave equation, or in the time-periodic case, by the Helmholtz equation. For problems with small wavelengths, high order discretizations must be used to resolve the solution. Two different techniques for finding compact finite difference schemes of high order are studied and compared. The first approach is Numerov's idea of using the equation to transfer higher derivatives to lower order ones for the Helmholtz equation, or, for the wave equation, from time to space. The second principle is the method of deferred correction, where a lower order approximation is used for error correction. For the time-independent Helmholtz problem, sharp estimates for the error are derived, in order to compare the arithmetic complexity for both approaches with a non-compact scheme. The characteristics of the errors for fourth order as well as sixth order accuracy are demonstrated and the advantages and disadvantages of the methods are discussed. A time compact, Numerov-type, fourth order method and a fourth order method using deferred correction in time are studied for the wave equation. Schemes are derived for both the second order formulation of the equation, and for the system in first order form. Stability properties are analyzed and numerical experiments have been performed, for both constant and variable coefficients in the equations. For the first order formulation, a staggered grid is used.