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Tsugio Fukuchi

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I am a researcher of numerical analysis.I retired from Fukushima prefecture 11 years ago.I am researching a high-accuracy, high-speed numerical calculation system using FDM (finite difference method).

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Michele Barbato

University of California, Davis

Mary Jane Curry

University of Rochester

Aly Mousaad Aly

Louisiana State University

University of Connecticut

Tiziana Lazzari

Università di Bologna

José Francisco Pessanha

UERJ - Universidade do Estado do Rio de Janeiro / Rio de Janeiro State University

The University of Manchester

Yildiz Technical University

Dragos Simandan

Brock University

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Papers by Tsugio Fukuchi

A comprehensive high-accuracy numerical calculation system for the 2D Poisson equation by the interpolation finite difference method

Research Squar, Preprint, 2023

To conduct numerical calculation in the finite difference method (FDM), a calculation system shou... more To conduct numerical calculation in the finite difference method (FDM), a calculation system should ideally be constructed to have three features: (i) the possibility of correspondence to an arbitrary boundary shape, (ii) high accuracy and (iii) high-speed calculation. In this study, the author has proposed and reported the interpolation FDM (IFDM) as a numerical calculation system with the above three characteristics. In this paper, we especially focus on (ii) high accuracy calculation. Regarding the 1D Poisson equation, the author has already reported on the overall picture of numerical calculations and proposed three schemes for high-accuracy numerical calculations: (i) the SAPI(m) scheme, (ii) SOBI(m) scheme, and (iii) CIFD(m) scheme. (m) denotes the accuracy order, which is usually an even number. Conventionally, high-order accuracy schemes up to the sixth order have been researched and reported, but theoretically, there is no upper accuracy order limit for (m) in these three s...

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

Research Square (Research Square), Jun 8, 2022

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic part... more 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 done on improving the accuracy of numerical calculations. Although there is much research in the field of FDM, the development of numerical calculations by the spectral method is decisive in improving 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 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 (CIFD()).

Accurate Empirical Calculation System for Predicting the Seepage Discharge and Free Surface Location of Earth Dam over Horizontal Impervious Foundation

Eng, Sep 26, 2020

In the numerical analysis of the Laplace equation, which is the governing equation of the seepage... more In the numerical analysis of the Laplace equation, which is the governing equation of the seepage phenomena of homogeneous, isotropic earth dams, it has been confirmed that numerical analysis with high accuracy is possible by using the interpolation finite difference method (IFDM). In a previous paper, based on this numerical analysis method, the equivalent Kozeny (KZ) flow method was proposed as a new empirical method to calculate the seepage discharges and free surface locations of earth dams. Although this method is generally a highly accurate method compared with the empirical method of A. Casagrande, owing to calculating the seepage problems within a few percentages of discharge relative errors, several additional studies are necessary. By integrating the finding of this study to the previous literature, an empirical seepage calculation system with high accuracy, the equivalent KZ flow method, is created. Owing to the finally proposed empirical method, called "interpolation-equivalent KZ flow method", the discharge and free surface location can be predicted with high accuracy in a wide range.

Interpolation numerical calculus for analytic functions by using algebraic polynomials

Research Square (Research Square), Mar 28, 2023

The finite difference method is a powerful method for numerical analysis of equilibrium and stead... more The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.

On Convergence Process of Turbulent Shear Flows in Ducts of Rectangular Cross-Section

Theoretical and applied mechanics Japan, 2003

High-accuracy and high-speed calculation of second-order ODE by the interpolation finite difference method

Research Square (Research Square), Aug 5, 2022

Numerical calculation of differential equations using the finite difference method (FDM) is enter... more Numerical calculation of differential equations using the finite difference method (FDM) is entering a new phase. The high-accuracy calculation system of the interpolation FDM (IFDM) enables extremely high-accuracy numerical calculation. One of the higher-order difference schemes of IFDM is defined as the compact interpolation finite difference (CIFD) scheme. According to this finite difference (FD) scheme, it becomes clear that it can be calculated with 15 significant figures in the numerical analysis of the 1D Poisson equation under double precision calculation. Thus, it is possible to perform an apparent error-free calculation. The 1D Poisson equation is a special form of the second-order ordinary differential equation (ODE), but this paper shows that the methods reported thus far are generalized to the general second-order ODE.

Numerical Analysis of Turbulent Flows in Ducts of Rectangular Cross-Section Using Harmonic Mixing Length

Theoretical and applied mechanics Japan, 2002

Numerical calculation of fully-developed laminar flows in arbitrary cross-sections using finite difference method

AIP Advances, Oct 6, 2011

The finite difference method has adequate accuracy to calculate fully-developed laminar flows in ... more The finite difference method has adequate accuracy to calculate fully-developed laminar flows in regular cross-sectional domains, but in irregular domains such flows are solved using the finite element method or structured grids. However, it has become apparent that we can use the finite difference method freely even if domains are complex. The non-slip condition on the wall must be imposed. Even in irregular domains, this boundary condition can be introduced indirectly by adding a single procedure to set the boundary condition. The calculations have similar accuracy as in regular domains. The proposed method has a wide range of applications; as a first step, fully-developed laminar flows are investigated in the paper.

High-Order Accurate and High-Speed Calculation System of 1D Laplace and Poisson Equations Using the Interpolation Finite Difference Method

Vide Leaf, Hyderabad eBooks, 2020

Among the methods of the numerical analysis of the physical phenomena of the continuum, the finit... more 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 highspeed computations become possible.

Interpolation numerical integration method for extremely high-accuracy calculation of ordinary differential equations

Research Square (Research Square), Nov 14, 2022

There are three types of problems that arise in the numerical analysis of differential equations:... more There are three types of problems that arise in the numerical analysis of differential equations: (i) the initial value problem (IVP), (ii) the boundary value problem (BVP), and (iii) IVP + BVP. In general, it is natural that (i) and (ii) are thoroughly examined before the study of (iii). Thus far, it has been shown that the BVP of one-dimensional (1D) Poisson equations can be solved with extremely high accuracy and speed using the interpolation finite difference method (IFDM). Furthermore, it has been found that a high-accuracy numerical calculation system for the IVP can be constructed by applying the polynomial interpolation method, as in the case of the BVP of the 1D Poisson equation. This method is called the interpolation numerical integration method (INIM). Numerical integration is possible with an arbitrary accuracy order by specifying the degree of the interpolation polynomial for representing the integrand. The proposed calculation method enables numerical calculations that seem to have no error. IVPs are usually calculated by the fourth-order accuracy Runge-Kutta method, which is often sufficient for engineering purposes. However, in this paper, we report that a numerical algorithm that agrees with the exact solution up to 15 significant digits has theoretical value.

Hydraulic Elements Chart for Pipe Flow Using New Definition of Hydraulic Radius

Journal of Hydraulic Engineering, Sep 1, 2006

The Manning formula is used routinely to calculate the mean velocity of uniform flow. Although th... more 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 c...

Characteristic features of error in high-order difference calculation of 1D Poisson equation and unlimited high-accurate calculation under multi-precision calculation

Mathematics and Computers in Simulation, Dec 1, 2021

Abstract In a previous paper based on the interpolation finite difference method, a calculation s... more Abstract In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

New high-precision empirical methods for predicting the seepage discharges and free surface locations of earth dams validated by numerical analyses using the IFDM

Soils and Foundations, Apr 1, 2018

The interpolation finite difference method (IFDM) can be used in the numerical analysis of steady... more The interpolation finite difference method (IFDM) can be used in the numerical analysis of steady-state seepage problems over arbitrary domains. In such analyses, by providing the coordinates of change points considering the polygonal domain and boundary conditions, a grid is automatically generated and a numerical solution is promptly obtained. Homogeneous, isotropic dams without tail drains have been investigated in a previous study. However, to consider dams with tail drains, we must alter the previous system slightly. This paper describes the altered system. In earlier times, when computation power was limited and before modern numerical calculation methods had been established, it was highly important for dam engineers to estimate the seepage discharge and the location of the free surface. Several empirical methods were proposed, and these have retained their functionality. For instance, the basic parabola method is an exemplary work that builds on classic empirical methods. In this work, the results obtained using A. Casagrande's method are compared with those obtained by IFDM, and it is confirmed that there are considerable deviations between them. The so-called ''equivalent KZ flow method" is proposed and shown to significantly improve the calculation accuracy for the discharge and free surface location.

Numerical stability analysis and rapid algorithm for calculations of fully developed laminar flow through ducts using time-marching method

AIP Advances, Mar 1, 2013

Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's... more

Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's equation over arbitrary domains

AIP Advances, Jun 1, 2014

Numerical analyses of steady-state seepage problems using the interpolation finite difference method

Soils and Foundations, Aug 1, 2016

Abstract Seepage analyses have mainly been executed using the finite element method; numerical an... more Abstract Seepage analyses have mainly been executed using the finite element method; numerical analyses using the finite difference method (FDM) have been limited to cases where the calculation domains are comparatively simple. This limitation is observed because FDM is considered to be inappropriate for application in calculations over complex domains. However, by applying the so-called “interpolation FDM (IFDM)”, we can now freely solve two- and three-dimensional elliptic partial differential equations (PDEs) over complex domains with high speed and high accuracy. By adopting this procedure, named the boundary polynomial interpolation, all of the numerical analyses of elliptic PDEs reduce to Dirichlet problems over regular domains. This method is also effective in the calculation of a flow net where mixed Dirichlet and Neumann conditions exist. By giving the coordinate values of changing points regarding the polygonal line of a domain and boundary conditions, grid generation is automatically carried out and numerical solutions are promptly obtained. In this paper, the method of saturated seepage analyses with a fixed domain is first formulated and then expanded to unconfined domain problems, namely, free surface problems. While analytical solutions of the PDE are highly limited, there is an analytical solution for the location of the free surface in a rectangular dam. The numerical solutions obtained using the IFDM are compared with the analytical ones, and it is shown that the proposed method has adequate accuracy in practice and wide applicability as a general method of numerically solving seepage problems.

Calculation error estimation without assuming a theoretical solution in the numerical calculation of the 1D Poisson equation by IFDM

Research Square (Research Square), Jun 17, 2022

The extreme high-accuracy calculation of the 1D Poisson equation by the interpolation finite diff... more The extreme high-accuracy calculation of the 1D Poisson equation by the interpolation finite difference method (IFDM) is possible. Numerical calculation errors have conventionally been evaluated by comparison with theoretical solutions. However, it is not always possible to obtain a theoretical solution. In addition, when trying to obtain numerical values from the theoretical solution, the exact numerical value may not be obtained because of inherent difficulties, that is, a theoretical solution equal to the exact numerical solution does not hold. In this paper, we focus on an ordered structure of the error calculated by the high-order accuracy finite difference (FD) scheme. This approach clarifies that in the numerical calculation of the 1D Poisson equation, which is the most basic ordinary differential equation, the error of the numerical calculation can be evaluated without comparison with the theoretical solution. Furthermore, in the numerical calculation by the IFDM, not only the discrete solution but also the continuous approximate solution is defined by piecewise interpolation polynomials.

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

AIP Advances, Oct 1, 2022

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic part... more 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)].

Interpolation numerical calculus for analytic functions by using algebraic polynomials

Interpolation numerical calculus for analytic functions by using algebraic polynomials

The finite difference method is a powerful method for numerical analysis of equilibrium and stead... more The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.

Interpolation numerical calculus for analytic functions by using algebraic polynomials

The finite difference method is a powerful method for numerical analysis of equilibrium and stead... more The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.

A comprehensive high-accuracy numerical calculation system for the 2D Poisson equation by the interpolation finite difference method

Research Squar, Preprint, 2023

To conduct numerical calculation in the finite difference method (FDM), a calculation system shou... more To conduct numerical calculation in the finite difference method (FDM), a calculation system should ideally be constructed to have three features: (i) the possibility of correspondence to an arbitrary boundary shape, (ii) high accuracy and (iii) high-speed calculation. In this study, the author has proposed and reported the interpolation FDM (IFDM) as a numerical calculation system with the above three characteristics. In this paper, we especially focus on (ii) high accuracy calculation. Regarding the 1D Poisson equation, the author has already reported on the overall picture of numerical calculations and proposed three schemes for high-accuracy numerical calculations: (i) the SAPI(m) scheme, (ii) SOBI(m) scheme, and (iii) CIFD(m) scheme. (m) denotes the accuracy order, which is usually an even number. Conventionally, high-order accuracy schemes up to the sixth order have been researched and reported, but theoretically, there is no upper accuracy order limit for (m) in these three s...

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

Research Square (Research Square), Jun 8, 2022

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic part... more 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 done on improving the accuracy of numerical calculations. Although there is much research in the field of FDM, the development of numerical calculations by the spectral method is decisive in improving 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 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 (CIFD()).

Accurate Empirical Calculation System for Predicting the Seepage Discharge and Free Surface Location of Earth Dam over Horizontal Impervious Foundation

Eng, Sep 26, 2020

In the numerical analysis of the Laplace equation, which is the governing equation of the seepage... more In the numerical analysis of the Laplace equation, which is the governing equation of the seepage phenomena of homogeneous, isotropic earth dams, it has been confirmed that numerical analysis with high accuracy is possible by using the interpolation finite difference method (IFDM). In a previous paper, based on this numerical analysis method, the equivalent Kozeny (KZ) flow method was proposed as a new empirical method to calculate the seepage discharges and free surface locations of earth dams. Although this method is generally a highly accurate method compared with the empirical method of A. Casagrande, owing to calculating the seepage problems within a few percentages of discharge relative errors, several additional studies are necessary. By integrating the finding of this study to the previous literature, an empirical seepage calculation system with high accuracy, the equivalent KZ flow method, is created. Owing to the finally proposed empirical method, called "interpolation-equivalent KZ flow method", the discharge and free surface location can be predicted with high accuracy in a wide range.

Interpolation numerical calculus for analytic functions by using algebraic polynomials

Research Square (Research Square), Mar 28, 2023

The finite difference method is a powerful method for numerical analysis of equilibrium and stead... more The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.

On Convergence Process of Turbulent Shear Flows in Ducts of Rectangular Cross-Section

Theoretical and applied mechanics Japan, 2003

High-accuracy and high-speed calculation of second-order ODE by the interpolation finite difference method

Research Square (Research Square), Aug 5, 2022

Numerical calculation of differential equations using the finite difference method (FDM) is enter... more Numerical calculation of differential equations using the finite difference method (FDM) is entering a new phase. The high-accuracy calculation system of the interpolation FDM (IFDM) enables extremely high-accuracy numerical calculation. One of the higher-order difference schemes of IFDM is defined as the compact interpolation finite difference (CIFD) scheme. According to this finite difference (FD) scheme, it becomes clear that it can be calculated with 15 significant figures in the numerical analysis of the 1D Poisson equation under double precision calculation. Thus, it is possible to perform an apparent error-free calculation. The 1D Poisson equation is a special form of the second-order ordinary differential equation (ODE), but this paper shows that the methods reported thus far are generalized to the general second-order ODE.

Numerical Analysis of Turbulent Flows in Ducts of Rectangular Cross-Section Using Harmonic Mixing Length

Theoretical and applied mechanics Japan, 2002

Numerical calculation of fully-developed laminar flows in arbitrary cross-sections using finite difference method

AIP Advances, Oct 6, 2011

The finite difference method has adequate accuracy to calculate fully-developed laminar flows in ... more The finite difference method has adequate accuracy to calculate fully-developed laminar flows in regular cross-sectional domains, but in irregular domains such flows are solved using the finite element method or structured grids. However, it has become apparent that we can use the finite difference method freely even if domains are complex. The non-slip condition on the wall must be imposed. Even in irregular domains, this boundary condition can be introduced indirectly by adding a single procedure to set the boundary condition. The calculations have similar accuracy as in regular domains. The proposed method has a wide range of applications; as a first step, fully-developed laminar flows are investigated in the paper.

High-Order Accurate and High-Speed Calculation System of 1D Laplace and Poisson Equations Using the Interpolation Finite Difference Method

Vide Leaf, Hyderabad eBooks, 2020

Among the methods of the numerical analysis of the physical phenomena of the continuum, the finit... more 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 highspeed computations become possible.

Interpolation numerical integration method for extremely high-accuracy calculation of ordinary differential equations

Research Square (Research Square), Nov 14, 2022

There are three types of problems that arise in the numerical analysis of differential equations:... more There are three types of problems that arise in the numerical analysis of differential equations: (i) the initial value problem (IVP), (ii) the boundary value problem (BVP), and (iii) IVP + BVP. In general, it is natural that (i) and (ii) are thoroughly examined before the study of (iii). Thus far, it has been shown that the BVP of one-dimensional (1D) Poisson equations can be solved with extremely high accuracy and speed using the interpolation finite difference method (IFDM). Furthermore, it has been found that a high-accuracy numerical calculation system for the IVP can be constructed by applying the polynomial interpolation method, as in the case of the BVP of the 1D Poisson equation. This method is called the interpolation numerical integration method (INIM). Numerical integration is possible with an arbitrary accuracy order by specifying the degree of the interpolation polynomial for representing the integrand. The proposed calculation method enables numerical calculations that seem to have no error. IVPs are usually calculated by the fourth-order accuracy Runge-Kutta method, which is often sufficient for engineering purposes. However, in this paper, we report that a numerical algorithm that agrees with the exact solution up to 15 significant digits has theoretical value.

Hydraulic Elements Chart for Pipe Flow Using New Definition of Hydraulic Radius

Journal of Hydraulic Engineering, Sep 1, 2006

The Manning formula is used routinely to calculate the mean velocity of uniform flow. Although th... more 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 c...

Characteristic features of error in high-order difference calculation of 1D Poisson equation and unlimited high-accurate calculation under multi-precision calculation

Mathematics and Computers in Simulation, Dec 1, 2021

Abstract In a previous paper based on the interpolation finite difference method, a calculation s... more Abstract In a previous paper based on the interpolation finite difference method, a calculation system was shown for calculating 1D (one-dimensional) Laplace’s equation and Poisson’s equation using high-order difference schemes. Finite difference schemes, from the usual second-order to tenth-order differences, including odd number order differences, were systematically and instantaneously derived over equally/unequally spaced grid points based on the Lagrange interpolation function. Using the direct method with the band diagonal matrix algorithm, 1D Poisson equations were numerically calculated under double precision floating arithmetic, but it became clear that high accurate calculations could not be secured in high-order differences because “digit-loss errors” caused by the finite precision of computations occurred in the calculations when using the high-order differences. The double precision calculation corresponds to 15 (significant) digit calculation. In this paper, we systematically investigate how the calculation accuracy changes by high precision calculations (30-digit, and 45-digit calculations). Under 45-digit calculation, where the digit-loss error can be almost ignored, the high-order differences enable extremely high-accurate calculations.

New high-precision empirical methods for predicting the seepage discharges and free surface locations of earth dams validated by numerical analyses using the IFDM

Soils and Foundations, Apr 1, 2018

The interpolation finite difference method (IFDM) can be used in the numerical analysis of steady... more The interpolation finite difference method (IFDM) can be used in the numerical analysis of steady-state seepage problems over arbitrary domains. In such analyses, by providing the coordinates of change points considering the polygonal domain and boundary conditions, a grid is automatically generated and a numerical solution is promptly obtained. Homogeneous, isotropic dams without tail drains have been investigated in a previous study. However, to consider dams with tail drains, we must alter the previous system slightly. This paper describes the altered system. In earlier times, when computation power was limited and before modern numerical calculation methods had been established, it was highly important for dam engineers to estimate the seepage discharge and the location of the free surface. Several empirical methods were proposed, and these have retained their functionality. For instance, the basic parabola method is an exemplary work that builds on classic empirical methods. In this work, the results obtained using A. Casagrande's method are compared with those obtained by IFDM, and it is confirmed that there are considerable deviations between them. The so-called ''equivalent KZ flow method" is proposed and shown to significantly improve the calculation accuracy for the discharge and free surface location.

Numerical stability analysis and rapid algorithm for calculations of fully developed laminar flow through ducts using time-marching method

AIP Advances, Mar 1, 2013

Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's... more

Finite difference method and algebraic polynomial interpolation for numerically solving Poisson's equation over arbitrary domains

AIP Advances, Jun 1, 2014

Numerical analyses of steady-state seepage problems using the interpolation finite difference method

Soils and Foundations, Aug 1, 2016

Abstract Seepage analyses have mainly been executed using the finite element method; numerical an... more Abstract Seepage analyses have mainly been executed using the finite element method; numerical analyses using the finite difference method (FDM) have been limited to cases where the calculation domains are comparatively simple. This limitation is observed because FDM is considered to be inappropriate for application in calculations over complex domains. However, by applying the so-called “interpolation FDM (IFDM)”, we can now freely solve two- and three-dimensional elliptic partial differential equations (PDEs) over complex domains with high speed and high accuracy. By adopting this procedure, named the boundary polynomial interpolation, all of the numerical analyses of elliptic PDEs reduce to Dirichlet problems over regular domains. This method is also effective in the calculation of a flow net where mixed Dirichlet and Neumann conditions exist. By giving the coordinate values of changing points regarding the polygonal line of a domain and boundary conditions, grid generation is automatically carried out and numerical solutions are promptly obtained. In this paper, the method of saturated seepage analyses with a fixed domain is first formulated and then expanded to unconfined domain problems, namely, free surface problems. While analytical solutions of the PDE are highly limited, there is an analytical solution for the location of the free surface in a rectangular dam. The numerical solutions obtained using the IFDM are compared with the analytical ones, and it is shown that the proposed method has adequate accuracy in practice and wide applicability as a general method of numerically solving seepage problems.

Calculation error estimation without assuming a theoretical solution in the numerical calculation of the 1D Poisson equation by IFDM

Research Square (Research Square), Jun 17, 2022

The extreme high-accuracy calculation of the 1D Poisson equation by the interpolation finite diff... more The extreme high-accuracy calculation of the 1D Poisson equation by the interpolation finite difference method (IFDM) is possible. Numerical calculation errors have conventionally been evaluated by comparison with theoretical solutions. However, it is not always possible to obtain a theoretical solution. In addition, when trying to obtain numerical values from the theoretical solution, the exact numerical value may not be obtained because of inherent difficulties, that is, a theoretical solution equal to the exact numerical solution does not hold. In this paper, we focus on an ordered structure of the error calculated by the high-order accuracy finite difference (FD) scheme. This approach clarifies that in the numerical calculation of the 1D Poisson equation, which is the most basic ordinary differential equation, the error of the numerical calculation can be evaluated without comparison with the theoretical solution. Furthermore, in the numerical calculation by the IFDM, not only the discrete solution but also the continuous approximate solution is defined by piecewise interpolation polynomials.

A whole high-accuracy numerical calculation system for the 1D Poisson equation by the interpolation finite difference method

AIP Advances, Oct 1, 2022

The interpolation finite difference method (IFDM) allows free numerical analysis of elliptic part... more 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)].

Interpolation numerical calculus for analytic functions by using algebraic polynomials

Interpolation numerical calculus for analytic functions by using algebraic polynomials

The finite difference method is a powerful method for numerical analysis of equilibrium and stead... more The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.

Interpolation numerical calculus for analytic functions by using algebraic polynomials

The finite difference method is a powerful method for numerical analysis of equilibrium and stead... more The finite difference method is a powerful method for numerical analysis of equilibrium and steady states of physical phenomena. In the traditional finite difference method, second-order accuracy differences have been used. From an engineering point of view, this approach is often regarded as having sufficient accuracy. However, much research has been done on increasing the accuracy of this numerical analysis. Virtual error zero (VE0) calculations are defined as calculations that can perform numerical calculations with exact solutions for 15 significant digits under double-precision calculations. VE0 calculation is considered to be the ultimate goal of high-accuracy numerical analysis. This paper is written from the viewpoint that such numerical calculations have inherent value. VE0 calculations are always obtained as long as we deal with functions having no singularities in the computational domain for ordinary differential equations as boundary value problems. Furthermore, VE0 calculation is possible even for ordinary differential equations as initial value problems. In both numerical analyses, algebraic polynomials commonly play an important role. This paper comprehensively examines the important role that algebraic polynomials play in increasing the accuracy of numerical calculations in various fields of numerical analysis, such as numerical differentiation, numerical integration, and numerical analysis of integral and integrodifferential equations.