Fourth-order Accuracy In Time Research Articles

Dispersion analysis of numerical wave propagation and its computational consequences

We present in this paper a comparison of the dispersion properties for several finite-difference approximations of the acoustic wave equation. We investigate the compact and staggered schemes of fourth order accuracy in space and of second order or fourth order accuracy in time. We derive the computational cost of the simulation implied by a precision criterion on the numerical simulation (maximum allowed error in phase or group velocity). We conclude that for moderate accuracy the staggered scheme of second order in time is more efficient, whereas for very precise simulation the compact scheme of fourth order in time is a better choice. The comparison increasingly favors the lower order staggered scheme as the dimension increases. In three dimensional simulation, the cost of extremely precise simulation with any of the schemes is very large, whereas for simulation of moderate precision the staggered scheme is the least expensive.

Extrapolation methods for dynamic partial differential equations

Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. One of these schemes reduces to a Runge-Kutta type formula when the equations are linear. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. The averaging procedure advocated by Gragg does not increase the efficiency of the scheme. For parabolic problems severe stability restrictions are encountered that limit the applicability to problems with large cell Reynolds number. Computational results are presented confirming the analytic discussions.