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The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic partial differential equations over arbitrary domains. Conventionally, in the finite difference method (FDM), the calculation is performed using the second-order accuracy central difference. For engineering problems, second-order accuracy calculations are often sufficient. On the other hand, much research has been carried out to improve the accuracy of numerical calculations. Although there is much research in the FDM field, the development of numerical calculations by the spectral method is decisive in improving the calculation accuracy. Numerical calculations are usually performed by double precision calculations. If double precision calculations ensure 15 significant digits in floating point computing, such numerical calculations will be the ultimate goal to reach. A numerical calculation that does not seem to have an error even though it originally has an error is defined as a virtual error-zero (VE0) calculation. In this paper, we will examine an overall picture of high-accuracy numerical calculation by the IFDM in the numerical calculation of the 1D Poisson equation. It becomes clear that a VE0 calculation is always possible in the numerical calculation method, defined as the compact interpolation finite difference scheme [(m)]. [ABSTRACT FROM AUTHOR]

The Manning formula is used routinely to calculate the mean velocity of uniform flow. Although this empirical formula is effective when applied to uniform flow in simple rectangular or trapezoidal cross sections, the roughness coefficient of the formula is variable when examining flow in a pipe that is partially full. Thus, the coefficient must be altered depending on the relative depth of fluid in the pipe. As this seems to be due to the definition of the hydraulic radius, a new definition of hydraulic radius is proposed here that was used to calculate a hydraulic elements chart for flow in pipes with a constant roughness coefficient. The results of the calculations showed very good agreement with Camp's chart. Furthermore, with adjustment of the 'free-surface weight factor,' this method was also capable of expressing other hydraulic elements charts reported previously. This new definition of hydraulic radius can also be applied to flow in simple cross sections and may be developed further for use with compound channel flows in future studies. DOI: 10.1061/(ASCE)0733-9429(2006)132:9(990) CE Database subject headings: Uniform flow; Pipe flow; Open channel flow; Sewer pipes; Velocity; Numerical models.

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. [ABSTRACT FROM AUTHOR]

Among the methods of the numerical analysis of the physical phenomena of the continuum, the finite difference method (FDM) is the first examined method and has been established as a full numerical calculation system over the regular domain. However, there is a general perception that generality in numerical calculations cannot be expected over complex irregular domains. As using the FDM, the development of computational methods that are applicable over any irregular domain is considered to be a very important contemporary problem. In the FDM, there is a marked characteristic that the theory developed by the (spatial) one-dimensional (1D) problem is naturally applied to the 2D and 3D problems. The calculation method is called the interpolation FDM (IFDM). In this paper, attention is paid to 1D Laplace and Poisson equations, and the whole image of the IFDM using the algebraic polynomial interpolation method (APIM), the IFDM-APIM, is described. Based on the Lagrange interpolation function, the spatial difference schemes from 2nd order to 10th order including odd order are calculated and defined instantaneously over equally/unequally spaced grid points, then, high-order accurate and high-speed computations become possible. [ABSTRACT FROM AUTHOR]