A Note on High-Precision Approximation of Asymptotically Decaying Solution and Orthogonal Decomposition

Abstract

In some physical applications, the decaying rate of asymptotically decaying solution is more important than the solution magnitude itself in understanding the physical system such as the late-time behavior of decaying fields in black hole space-time. In Khanna (J Sci Comput 56(2):366–380, 2013), it was emphasized that high-precision arithmetic and high-order methods are required to capture numerically the correct decaying rate of the late-time radiative tails of black-hole system in order to prevent roundoff errors from inducing a wrong power-law decay rate in the numerical approximation. In this paper, we explain how roundoff errors induce a wrong decay mode in the numerical approximation using simple linear differential equations. Then we describe the orthogonal decomposition method as a possible technique to remove wrong decaying modes induced by roundoff errors in the numerical approximation.