Numerical Anisotropy in Finite Differencing
For computational scientists solving multi-dimensional wave propagation problems, this review consolidates knowledge on numerical anisotropy and optimization strategies, though it is a survey of existing work.
This review describes numerical anisotropy, a discretization error causing direction-dependent wave speeds in multi-dimensional finite difference solutions of hyperbolic PDEs, and discusses studies optimizing schemes to reduce it.
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 multi-dimensions, where the waves propagate in all directions, there is an additional specific error resulting from the discretization of spatial derivatives along grid lines. Specifically, waves or wave packets in multi-dimensions propagate at different phase or group velocities, respectively, along different directions. A commonly used term for the aforementioned multidimensional 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 multi-dimensions. Then, several important studies that were focused on optimizations of finite difference schemes with the objective of reducing the numerical anisotropy are discussed.