Implementation of One and Two-Step Generalized Integral Representation Methods (GIRMs)
Hideyuki Niizato^{1}, Gantulga Tsedendorj^{2}, Hiroshi Isshiki^{3}
^{1}Hitachi Zosen Corporation, Osaka, Japan
^{2}Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia
^{3}IMA, Institute of Mathematical Analysis, Osaka, Japan
Email address:
(H. Niizato), (G. Tsedendorj), (H. Isshiki)
To cite this article:
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
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
Add to Repeat process. (30)
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
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