Infinite-order accuracy limit of finite difference formulas in the complex plane

Abstract

It was recently found that finite difference (FD) formulas become remarkably accurate when approximating derivatives of analytic functions $f(z)$ in the complex $z=x+\text{i}y$ plane. On unit-spaced grids in the $x,y$-plane, the FD weights decrease to zero with the distance to the stencil center at a rate similar to that of a Gaussian, typically falling below the level of double precision accuracy $\mathcal{O}(10^{-16})$ already about four node spacings away from the center point. We follow up on these observations here by analyzing and illustrating the features of such FD stencils in their infinite-order accurate limit (for traditional FD approximations known as their pseudospectral limit).

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

“…is uniquely defined, no matter from which direction in the complex plane ∆x 1 approaches zero [17,18]. Any function u(x) that possesses a Taylor expansion at some x-location can be extended to an analytic function, with a vast range of further consequences [17][18][19][20].…”

“…is uniquely defined, no matter from which direction in the complex plane ∆x 1 approaches zero [17,18]. Any function u(x) that possesses a Taylor expansion at some x-location can be extended to an analytic function, with a vast range of further consequences [17][18][19][20]. Now, instead of having forward, backward, and central difference approximations on the real line, we have first quadrant (Q1), second quadrant (Q2), third quadrant (Q3), fourth quadrant (Q4), and central (C) on the complex plane.…”

“…Finite difference formulas in the complex plane are in general more accurate than finite difference formulas in the real domain [17][18][19][20]. The reasons for this are that complex derivatives depend equally on data from all directions around the point, i.e., stencils must not extend very far, only along a line, and the Cauchy-Riemann equations impose restrictions on the range of functions that need to be considered.…”