Abstract
In a previous paper, a calculation system for a high-accuracy, high-speed calculation of a one-dimensional (1D) Poisson equation based on the interpolation finite difference method was shown. Spatial high-order finite difference (FD) schemes, including a usual second-order accurate centered space FD scheme, are instantaneously derived on the equally spaced/unequally spaced grid points based on the definition of the Lagrange polynomial function. The upper limit of the higher order FD scheme is not theoretically limited but is studied up to the tenth order, following the previous paper. In the numerical analyses of the 1D Poisson equation published in the previous paper, the FD scheme setting method, SAPI (m), m = 2, 4, …, 10, was defined. Due to specifying the value of m, the setting of FD schemes is uniquely defined. This concept is extended to the numerical analysis of two-dimensional Poisson equations. In this paper, we focus on Poiseuille flows passing through arbitrary cross sections as numerical calculation examples. Over regular and irregular domains, three types of FD methods—(i) forward time explicit method, (ii) time marching successive displacement method, and (iii) alternative direction implicit method—are formulated, and their characteristics of convergence and numerical calculation errors are investigated. The numerical calculation system of the 2D Poisson equation formulated in this paper enables high-accuracy and high-speed calculation by the high-order difference in an arbitrary domain. Especially in the alternative direction implicit method using the band diagonal matrix algorithm, convergence is remarkably accelerated, and high-speed calculation becomes possible.
Highlights
Partial differential equations (PDEs) are classified into (i) parabolic PDE, (ii) elliptic PDE, and (iii) hyperbolic PDE.1,2. All these three PDEs are expressed by finite difference equations (FDEs) and numerically calculated, they are common in handling spatial finite differences. (i) The parabolic PDE expresses how a certain physical phenomenon changes over time in a predetermined domain, and it converges to a certain steady state under calculation conditions in which the calculation parameters do not depend on the passage of time
The finite difference method (FDM) is the first and most used method for numerical analysis of PDEs that represent the physical phenomena of a continuum
The interpolation finite difference method (IFDM) system proposed by the author28–33 is considered to be one such study
Summary
Partial differential equations (PDEs) are classified into (i) parabolic PDE, (ii) elliptic PDE, and (iii) hyperbolic PDE. all these three PDEs are expressed by finite difference equations (FDEs) and numerically calculated, they are common in handling spatial finite differences. (i) The parabolic PDE expresses how a certain physical phenomenon changes over time in a predetermined domain, and it converges to a certain steady state under calculation conditions in which the calculation parameters do not depend on the passage of time. We have already mentioned that the HOC-FD scheme for highaccuracy calculations has been proposed, but in this scheme, the forcing term f (x, y) of the Poisson equation (described later) is assumed to be sufficiently smooth and to have the required continuous partial derivatives This method is powerless in the numerical calculation of the Poiseuille flow examined in this paper.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
