Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation

A time‐dependent nonlocal wave equation is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The Newmark scheme is considered for discretizing the time derivative and piecewise‐linear finite element methods are used for spatial discretization. For certain ranges of a parameter appearing in the Newmark scheme, unconditional stability is proved; in particular, this result applies to the backward‐Euler‐like and Crank‐Nicolson‐like schemes. For other values of the parameter which includes the forward‐Euler‐like scheme, conditional stability is proved. Dispersion relations for the nonlocal wave equation in one and two dimensions are derived. Comparisons with the analogous results for the classical wave equation are provided as the results of numerical experiments that illustrate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 500–516, 2015