Numerical study of unsteady diffusion in circle

G. Tsedendorj; H. Isshiki

doi:10.1007/s40314-019-0794-8

Computational and Applied Mathematics

March 2019, 38:26 | Cite as

Numerical study of unsteady diffusion in circle

Steady-state fundamental solution approach

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Authors and affiliations

G. Tsedendorj
H. Isshiki

Article

First Online: 23 January 2019

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Abstract

In this study, we present a discretization scheme based on generalized integral representation method for a numerical evaluation of an unsteady diffusion problem in a circular domain. The scheme employs the fundamental solution of the associated steady-state diffusion operator along with piecewise constant approximation for the unknown function. By its construction, our scheme does not require continuity of the approximate solution across the computational elements and thus is flexible for various partitions of the problem domain. Therefore, for numerical validation, we provide examples in the unit circle partitioned via three different manners. Examples on triangulated partitions of the unit square are also included. The derivation of the numerical scheme is straightforward and it is easy-to-program.

Keywords

Numerical solution of diffusion problem Unsteady diffusion in a circle Steady-state fundamental solution Generalized integral representation method (GIRM) Numerical scheme based on GIRM

Mathematics Subject Classification

65N99 65L99

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Notes

Acknowledgements

The authors thank the anonymous reviewer for his comments and suggestions, which helped to improving the quality of the manuscript. The authors thank E. Avirmed for the help of code debugging and conducting tests. This work has been done within the framework of the Project P2017-2412 supported by the National University of Mongolia.

Appendix

Analytical solutions of test problems

Below we provide the analytical solutions of the test problems used in Sects. 4.1 and 4.2 in their series expansions.

Analytical solutions in the unit circle

The analytical solutions of the problems used in Test 1 are

$$\begin{aligned} \begin{aligned} C(r,t)&= \sum _{k=1}^{\infty } \frac{2}{j_{0,k}J_1(j_{0,k})} \exp \left( -{\kappa j^2_{0,k} t} \right) J_0 \left( j_{0,k} {r} \right) \text { for } f_1(\mathbf {x}) \\ C(r,t)&= \sum _{k=1}^{\infty } \frac{16 \sin (\theta )}{j_{1,k}^3 J_2(j_{1,k})} \exp \left( -{\kappa j^2_{1,k} t} \right) J_1 \left( j_{1,k} {r} \right) \text { for } f_2(\mathbf {x}), \end{aligned} \end{aligned}$$

(13)

where $r=\sqrt{x^2+y^2}$, $\theta =\tan ^{-1} \left( \frac{y}{x} \right) $; $J_\alpha (x)$ is Bessel function of the first kind of order $\alpha = 0,1,2 $ and $j_{\alpha ,k}, k=1,...$ are zeros of $J_{\alpha }(x)$.

Analytical solutions in the unit square

The analytical solutions of the problems used in Test 2 are expressed in double series expansions as:

$$\begin{aligned} \begin{aligned} C(\mathbf {x},t)&=\frac{16}{\pi ^2}\sum _{n=1}^{\infty } \sum _{m=1}^{\infty } \exp \left( -(k^2+l^2) \pi ^2 t \right) A_{kl}(x,y) \text { for } f_1(\mathbf {x}) \\ C(\mathbf {x},t)&=\frac{64}{\pi ^4}\sum _{n=1}^{\infty } \sum _{m=1}^{\infty } (-1)^{n+m} \exp \left( -(k^2+l^2)\pi ^2 t \right) \frac{A_{kl}(x,y)}{k l} \text { for } f_2(\mathbf {x}), \end{aligned} \end{aligned}$$

(14)

with $k=2n-1$, $l=2m-1$ and $A_{kl}(x,y)={\sin (k \pi x) \sin (l \pi y)}/{(kl)}$.

Analytical formulae for singular integrals

The integrals in $\mathbf {A}$ and $\mathbf {D}$ in Eq. (10) can be expressed by the following formulae at their points of singularities.

Analytical formula for boundary integral

In the case when boundary element ${\Gamma }_m$ is a straight line segment, it can be shown that the boundary integral in $\mathbf {D}$ in Eq. (10) is expressed by

$$\begin{aligned} \int _{\Gamma _m} \ln \left| \mathbf {\upxi }^b - \mathbf {x}^b_j \right| d{\Gamma _m} = a \ln a + b \ln b -a -b, \end{aligned}$$

(15)

if $\mathbf {\upxi }^b=\mathbf {x}^b_j$. Here, a and b are the distances from $\mathbf {\upxi }^b$ to the end points of ${\Gamma }_m$ as shown in Fig. 4a.

Analytical formula for domain integral

In the case of interior element ${\Omega }_n$ being a convex polygon, the domain integral in $\mathbf {A}$ at its singular point $\mathbf {\upxi }=\mathbf {x}_i$, is analytically expressed as follows. The element ${\Omega }_n$ is divided into p subtriangles $\Delta _1$, $\Delta _2$, $\dots $, $\Delta _p$ by joining the point $\mathbf {\upxi }$ to the vertices, where $p \ge 3$ is the number of vertices. Integrating around $\mathbf {\upxi }$ by angle $\theta $ on each subtriangle and adding up the results of separate integrations, we obtain

$$\begin{aligned} \iint _{\Omega _n} \ln \left| \mathbf {\upxi } - \mathbf {x}_i \right| d{\Omega _n} = \sum _{\Delta _1, \dots , \Delta _p} \frac{c}{2} \sin \beta \left[ \ln \left( \tan \frac{\beta }{2} \right) - \ln \left( \tan \frac{\alpha + \beta }{2} \right) \right] , \end{aligned}$$

(16)

where $\alpha $, $\beta $ and c are angles and side length of subtriangle $\Delta _k$, $k=1, 2, \dots , p$ as shown in Fig. 4b.

At its ordinary points, i.e. when $\mathbf {\upxi } \ne \mathbf {x}_i$, the integral in $\mathbf {A}$ on ${\Omega }_n$ is approximated by the sum of separate integrals on $\Delta _k$, $k=1, 2, \dots , p$. Each separate integral in turn is computed via 3-point Gauss–Legendre quadrature.

Equation (16) is an apparent extension of the analytical formula presented in Kirkup (2007). It holds for any interior element of MP, AT and EA-like meshes used in Test 1 and Test 2 in Sect. 4.1 and 4.2, respectively.

References

Al-Jawary MA, Ravnik J, Wrobel LC, Skerget L (2012) Boundary element formulations for the numerical solution of two-dimensional diffusion problems with variable coefficients. Comput Math Appl 64:2695–2711MathSciNetCrossRefGoogle Scholar
Beckers B, Beckers P (2012) A general rule for disk and hemisphere partition into equal-area cells. Comput Geom 45:275–283MathSciNetCrossRefGoogle Scholar
Chantasiriwan S (2006) Methods of fundamental solutions for time-dependent heat conduction problems. Int J Numer Methods Eng 66:147–165CrossRefGoogle Scholar
Chen L, Liew KM (2011) A local Petrov–Galerkin approach with moving kriging interpolation for solving transient heat conduction problems. Comput Mech 47:455–467MathSciNetCrossRefGoogle Scholar
Chen W, Wang F (2016) Singular boundary method using time-dependent fundamental solution for transient diffusion problems. Eng Anal Bound Elem 68:115–123MathSciNetCrossRefGoogle Scholar
Chen CS, Cho HA, Golberg MA (2006) Some comments on the Ill-conditioning of the method of fundamental solutions. Eng Anal Bound Elem 30:405–410CrossRefGoogle Scholar
Feng W-Z, Yang K, Cui M, Gao X-W (2016) Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems. Int Commun Heat Mass 79:21–30CrossRefGoogle Scholar
Hong Y (2015) Numerical approximation of the singularly perturbed heat equation in a circle. J Sci Comput 62:1–24MathSciNetCrossRefGoogle Scholar
Isshiki H (2015) Effects of generalized fundamental solution on generalized integral representation method. Appl Comput Math 4:40–51CrossRefGoogle Scholar
Isshiki H (2015) From integral representation method (IRM) to generalized integral representation method (GIRM). Appl Comput Math 4(3–1):1–14CrossRefGoogle Scholar
Jayantha PA, Turner IW (2001) A comparison of gradient approximations for use in finite-volume computational models for two-dimensional diffusion equations. Numer Heat Transf B Fund 40:367–390CrossRefGoogle Scholar
Johansson BT, Lesnic D, Reeve T (2011) A method of fundamental solutions for two-dimensional heat conduction. Int J Comput Math 88:1697–1713MathSciNetCrossRefGoogle Scholar
Kirkup S (2007) The boundary element method in acoustics, 2nd edn. Integrated Sound Software. http://www.cad-cam-cae.com/blog/Kirkup07TBEMIA.htm
Kolodziej JA, Mierzwiczak M (2010) Transient heat conduction by different versions of the method of fundamental solutions—a comparison study. Comput Assist Mech Eng Sci 17:75–88MathSciNetGoogle Scholar
Kouatchou J (2003) Comparison of time and spatial collocation methods for the heat equation. J Comput Appl Math 150:129–141MathSciNetCrossRefGoogle Scholar
Lai MC, Tseng YH (2005) A fast iterative solver for the variable coefficient diffusion equation on a disk. J Comput Phys 208:196–205MathSciNetCrossRefGoogle Scholar
Liu H, Yan J (2009) The direct discontinuous Galerkin methods for diffusion problems. SIAM J Numer Anal 47:675–698MathSciNetCrossRefGoogle Scholar
Lorcher F, Gassner G, Munz C-D (2008) An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J Comput Phys 227:5649–5670MathSciNetCrossRefGoogle Scholar
Mai W, Soghrati S, Buchheit R (2016) A Green’s discrete transformation meshfree method for simulating transient diffusion problems. Int J Numer Methods Eng 108:252–270MathSciNetCrossRefGoogle Scholar
Persson P-O, Strang G (2004) A simple mesh generator in MATLAB. SIAM Rev 46:329–345MathSciNetCrossRefGoogle Scholar
Ramachandran PA (2002) Method of fundamental solutions: singular value decomposition analysis. Commun Numer Methods Eng 18:789–801CrossRefGoogle Scholar
Shampine LF (2008) MATLAB program for quadrature in 2D. Appl Math Comput 202:266–274MathSciNetzbMATHGoogle Scholar
Shampine LF (2008) Vectorized adaptive quadrature in MATLAB. J Comput Appl Math 211:131–140MathSciNetCrossRefGoogle Scholar
Tsedendorj G, Isshiki H (2017) Numerical solution of two-dimensional advection–diffusion equation using generalized integral representation method. Int J Comput Methods 14:1750028 (1–13) MathSciNetCrossRefGoogle Scholar
Wang H, Qin QH (2009) Hybrid FEM with fundamental solutions as trial functions for heat conduction simulation. Acta Mech Solida Sin 22:487–498CrossRefGoogle Scholar
Young DL, Tsa CC, Murugesan K, Fan CM, Chen CW (2004) Time-dependent fundamental solutions for homogeneous diffusion problems. Eng Anal Bound Elem 28:1463–1473CrossRefGoogle Scholar

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Authors and Affiliations

G. Tsedendorj
- 1
Email authorView author's OrcID profile
H. Isshiki
- 2

1.Department of Mathematics, School of SciencesNational University of MongoliaUlaanbaatarMongolia
2.Institute of Mathematical AnalysisOsakaJapan

Numerical study of unsteady diffusion in circle

Abstract

Keywords

Mathematics Subject Classification

Notes

Acknowledgements

References

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