| We have investigated the very high computational efficiency of high-order finite difference methods, especially as they incorporate features such as implicitness (also known as compactness in the literature) and grid staggering. While remaining relatively compact, these methods can approach the superior accuracy and effectiveness of spectral methods while still allowing some boundary flexibility. In the past, grid staggering has been observed to be beneficial in some cases (e.g. the Yee scheme for computational electrodynamics), but that idea has been shown here to combine favorably with both implicitness and high orders of accuracy.; In addition, we have explored the new idea of grid staggering for time integrators. In the important application of solving linear wave equations (e.g. acoustic or elastic waves equations, or Maxwell's equations for electromagnetic fields), nearly an order of magnitude gain can usually be achieved in accuracy (for the same computational cost in both operation count and in memory) compared to classical ODE solvers such as Adams or Runge-Kutta methods. In addition, our new staggered methods have superior stability properties to the classical methods in the context of solving wave equations. We investigate the accuracy and stability of these methods analytically, experimentally, and through the use of a novel root portrait technique. We also consider several theoretical questions concerning these staggered time integrators. |
