Applied and Computational Mathematics
Volume 4, Issue 3-1, June 2015, Pages: 59-77

Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs)

Hideyuki Niizato1, Gantulga Tsedendorj2, Hiroshi Isshiki3

1Hitachi Zosen Corporation, Osaka, Japan

2Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia

3IMA, Institute of Mathematical Analysis, Osaka, Japan

(H. Niizato), (G. Tsedendorj), (H. Isshiki)

Hideyuki Niizato, Gantulga Tsedendorj, Hiroshi Isshiki. Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs). Applied and Computational Mathematics. Special Issue: Integral Representation Method and its Generalization. Vol. 4, No. 3-1, 2015, pp. 59-77. doi: 10.11648/j.acm.s.2015040301.15

Abstract: In this study, we summarize and implement one- and two-step Generalized Integral Representation Methods (GIRMs). Although GIRM requires matrix inversion, the solution is stable and the accuracy is high. Moreover, it can be applied to an irregular mesh. In order to validate the theory, we apply one- and two-step GIRMs to the one-dimensional Initial and Boundary Value Problem for advective diffusion. The numerical experiments are conducted and the approximate solutions coincide with the exact ones in both cases. The corresponding computer codes implemented in most popular computational languages are also given.

Keywords: Initial and Boundary Value Problem (IBVP), Generalized Fundamental Solution, Generalized Integral Representation Method (GIRM), Implementation of GIRM, Computer Codes

1. Introduction

Integral Representation Method (IRM) and its generalization - Generalized Integral Representation Method (GIRM) are one of the most convenient methods to numerically solve Initial and Boundary Value Problems (IBVP) such as advection-diffusion type equations. Both methods can be applied to an irregular mesh, and the solution is stable and accurate. In Ref. [1], generalization from IRM to GIRM is discussed not only from the theoretical viewpoint, but also from the computational aspects. Comparison and relationship with other numerical methods are also given. As an example, GIRM is applied to one- and two-dimensional diffusion problems and two-dimensional Burgers’ equation. Moreover, it is also shown that the determination of a fundamental solution for GIRM is always possible in advance.

As another demonstration, GIRM is applied to fluid dynamic motion of gas or particles to obtain the accurate numerical solutions [2]. The numerical results by the GIRM are compared with the solutions by Finite Difference Method (FDM). The GIRM produces reasonable and accurate numerical solutions.

In the present study, a brief summary of the theory of One- and Two-step Generalized Integral Representation Methods (GIRMs) are presented. To validate the theory, numerical experiments of the one-dimensional IBVP for advective diffusion are conducted. Highly accurate numerical results are obtained in both cases of One- and Two-step GIRMs in admissible time periods. The corresponding computer codes implemented in widely used programming languages such as Matlab, C and FORTRAN are also given.

2. One-Step Generalized Integral Representation Method (1-Step GIRM)

2.1. Summary of Theory

We discuss numerical solutions of the IBVP for advective diffusion. For simplicity, we apply 1-step GIRM to the one-dimensional problem with constant advection velocity and constant diffusion coefficient.

If  is substance density in the region , the corresponding IBVP is given by

Differential equation:

(1)

Boundary condition:

(2)

Initial condition:

(3)

where  is space coordinate,  is time,  is advection velocity,  is diffusion constant and  is source of the substance, respectively.

Multiplying both sides of Eq. (1) by a function  of  and  and integrating over , we obtain

(4)

If we rewrite Eq. (4) and exchange  and , we have

(5)

where  is a Generalized Fundamental Solution (GFS) chosen properly. For instance, we take Gaussian GFS:

(6)

Eq. (5) is a generalized integral representation of Eq. (1). This integral representation is applied to numerical solution. If  in ,  and  are known, then Eq. (5) is an integral equation with unknowns  in ,  and , where  is the kernel function. Namely, we are able to obtain  numerically, if we use for instance, the following procedure:

Let  be known at time

Obtain  from Eq. (5)

Apply

Increment time by  Repeat process.     (7)

2.2. Numerical Experiment

To begin with, we introduce for example, a regular mesh:

,

(8)

(9)

and denote

(10)

We prepare the following approximations for discretization of Eq. (5)

(11a)

(11b)

(11c)

(11d)

where  etc.

Thus Eq. (5) can be discretized

(12)

where

(13)

The unknowns in Eq. (12) are , ,  and . Eq. (12) is satisfied at the interior points  as well as at the boundary points . Hence, we have  equations for  unknowns. Furthermore, if we use approximations

,

(14)

and satisfy Eq. (12) at the interior points, we have  equations for  unknowns.

Although GIRM is computationally complex and requires matrix inversion, its accuracy of the numerical results is high. It can be applied to an irregular mesh. If a computer code is properly composed and implemented, the computational load might be comparable with the Finite element Method (FEM).

Numerical examples of the GIRM are given below. The initial condition is exponential and given by

(15)

We assume that  is large enough, and the boundary condition is specified as

(16)

The exact solution is given by

(17)

In order to reduce spurious oscillation in numerical solutions, it is effective to use a finer mesh, but it invites serious increase of computational resource. Therefore, if necessary, numerical damping

(18)

is added to  at every time step, where  is a damping constant. Moreover, if the discontinuity of the initial density causes serious numerical errors, it is effective to replace  with a filtered value such as

(19)

Finally, for the reduction of computation time, numerical integrals that include function G and its derivatives w.r.t  in the right hand side of Eq. (5) are calculated in the neighborhood of  only:

(20)

(a) ∂C/∂t

(b) ∂C/∂x

(c) C

Figure 1. Numerical solutions by 1-Step GIRM: (a) Time derivative ∂C/∂t, (b) Space derivative ∂C/∂x and (c) Solution C itself, where used exponential initial density distribution.

Numerical results are shown in Fig. 1. The accuracy of the numerical results is very high and it coincides with the exact ones. Values of the parameters used in the numerical experiment are:

(21)

The computer codes are given in Appendix A.

3. Two-Step Generalized Integral Representation Method (2-Step GIRM)

3.1. Summary of Theory

In order to derive two-step GIRM for Eq. (1), we rewrite it as follows:

Non-uniformity equation:        (22)

Constitutive equation:          (23)

Equilibrium equation:     (24)

Multiplying both sides of Eq. (22) by function  and integrating over , we obtain

(25)

where

(26)

Rewriting Eq. (25) and exchanging  and , we obtain a generalized integral representation for Eq. (22):

(27)

A generalized integral representation of Eq. (24) is obtained similarly:

(28)

Rewriting Eq. (28) and exchanging  and , we obtain a generalized integral representation for Eq. (24):

(29)

Hence, we are able to solve  numerically, if we use for example, the following procedure:

Let  be known  Obtain  from (27)

Obtain  from (23)  Obtain  from (29)

Then

3.2. Numerical Experiment

Numerical examples are given for the same problem as in case of the 1-Step GIRM. The initial and boundary conditions are given by Eq. (3) and Eq. (16), respectively. The exact solution is given by Eq. (17). Values of the parameters used in the numerical experiment are:

(31)

Numerical results are shown Fig. 2. The accuracy of the numerical results is very high and it coincides with the exact ones. The corresponding computer codes are given in Appendix B.

4. Characteristics of Generalized Fundamental Solutions (GFS)

In the numerical experiments in 2.2 and 3.2, Gaussian GFS was used. However, Gaussian GFS causes some problems, when the boundary value is not zero [3,4]. The remedies are given in Ref. [3,4]. For example,  GFS such as harmonic and exponential GFSs works well. If the non-zero boundary value problem is transformed into a zero boundary value problem, Gaussian GFS is very effective.

(a) ∂C/∂t

(b) ∂C/∂x

(c) C

Figure 2. Numerical solutions by 2-Step GIRM: (a) Time derivative ∂C/∂t, (b) Space derivative ∂C/∂x and (c) Solution C itself, where used exponential initial density distribution.

5. Conclusion

GIRM is a convenient alternative to numerically solve the IBVP such as advective diffusion. Numerical solutions obtained by GIRM in particular, by 1- and 2-step GIRMs are stable and accurate.

In the present paper, we summarize and implement 1- and 2-step GIRMs. As implementation demonstration, we provide computer codes written in widely used computational languages including low-level programming languages such as C and FORTRAN as well as high-level language such as Matlab.

Appendix A. Computer codes of 1-Step GIRM

A.1. Matlab code

A.2. C code

A.3. FORTRAN code

Appendix B. Computer codes of 2-Step GIRM

B.1. Matlab code

B.2. C code

B.3. FORTRAN code

References

1. H. Isshiki,"From Integral Representation Method (IRM) toGeneralized Integral Representation Method (GIRM),"Applied and ComputationalMathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication. http://www.sciencepublishinggroup.com/ journal/archive.aspx?journalid=147&issueid=-1
2. H. Isshiki,T. Takiya, and H. Niizato, "Application of Generalized Integral representation (GIRM) Method to Fluid DynamicMotion of Gas or Particles in Cosmic Space Driven by Gravitational Force,"Applied and ComputationalMathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication. http://www.sciencepublishinggroup.com/journal/archive.aspx?journalid=147&issueid=-1
3. H. Isshiki, "Effects of Generalized Fundamental Solution (GFS) onGeneralized Integral Representation Method (GIRM)," Applied and ComputationalMathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication.http://www.sciencepublishinggro up .com/journal/archive.aspx?journalid=147&issueid=-1
4. H. Isshiki, "Application of the Generalized Integral Representation Method (GIRM) to Tidal Wave Propagation," Applied and ComputationalMathematics, Special Issue: Integral Representation Method and Its Generalization, (2015), under publication.http://www.sciencepublishinggroup.com/ journal/archive.aspx?journalid=147&issueid=-1

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