Numerical anisotropy in finite differencing

• Published in: Advances in difference equations Vol. 2015; no. 1; pp. 1 - 17
• Main Author: Sescu, Adrian
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 16.01.2015; BioMed Central Ltd
• Subjects: Analysis; Anisotropy; Difference and Functional Equations; Discretization; Dispersions; Errors; Functional Analysis; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Review; Wave equations; Wave propagation; Advection Equation; Numerical Dispersion; Helmholtz Equation; Finite Difference Scheme; Compact Scheme
• ISSN: 1687-1847; 1687-1847
• DOI: 10.1186/s13662-014-0343-0

Numerical solutions to hyperbolic partial differential equations, involving wave propagations in one direction, are subject to several specific errors, such as numerical dispersion, dissipation or aliasing. In the multi-dimensional case, where the waves propagate in all directions, there is an additional specific error resulting from the discretization of spatial derivatives along the grid lines. Specifically, waves or wave packets in the multi-dimensional case propagate at different phase or group velocities, respectively, along different directions. A commonly used term for the aforementioned multi-dimensional discretization error is the numerical anisotropy or isotropy error. In this review, the numerical anisotropy is briefly described in the context of the wave equation in the multi-dimensional case. Then several important studies that were focused on optimizations of finite difference schemes with the objective of reducing the numerical anisotropy are discussed.