- Research
- Open Access
Unique solvability of the CCD scheme for convection–diffusion equations with variable convection coefficients
- Qinghe Wang^{1},
- Kejia Pan^{1} and
- Hongling Hu^{2}Email author
https://doi.org/10.1186/s13662-018-1591-1
© The Author(s) 2018
- Received: 27 December 2017
- Accepted: 5 April 2018
- Published: 4 May 2018
Abstract
The combined compact difference (CCD) scheme has better spectral resolution than many other existing compact or noncompact high-order schemes, and is widely used to solve many differential equations. However, due to its implicit nature, very little theoretical results on the CCD method are known. In this paper, we provide a rigorous theoretical proof for the unique solvability of the CCD scheme for solving the convection-diffusion equation with variable convection coefficients subject to periodic boundary conditions.
Keywords
- Combined compact difference scheme
- Convection-diffusion equation
- Unique solvability
- Variable coefficient
- Periodic boundary conditions
MSC
- 65M06
- 65M32
1 Introduction
In many real physical applications, performing high-order and efficient numerical methods for solving the partial differential equations is essential. In particular, it is important to simultaneously solve the unknown function and its derivatives with high-order accuracy. For example, Lele [1] has shown that when the schemes involve not only the value of the function but also those of its derivatives, spectral-like resolution can be achieved while keeping a small stencil. Many attempts have been made to develop such schemes involving both the unknown function and its derivatives. Among these methods, the three-point six-order combined compact difference scheme (CCD) proposed by Chu and Fan [2–4] is well known and popular for its high efficiency. The CCD scheme, which can be regarded as an extension of the standard Pade schemes as discussed by Lele [1], allows us to conveniently handle the differential equations with variable coefficients subject to Robin boundary conditions. When using the CCD method to solve the differential equations, the equation is assumed to be valid at the boundary, and the first and second derivatives together with the function values of unknowns at grid points are computed simultaneously [2]. Fourier analysis shows that the CCD scheme is more accurate than many other compact or noncompact schemes [2]. Since its appearance, there has been a lot of research discussing the application and improvement of the method [5–15].
The CCD method is originally proposed to solve second-order linear ordinary differential equations [2]. For multi-dimensional evolution problems, we can employ alternating direction implicit (ADI) technique to convert it into a series of one-dimensional (1D) problems, which can be solved efficiently by the CCD scheme [16–22]. Lee et al. [23] developed a CCD method for directly solving the general two-dimensional (2D) linear partial differential equation with a mixed derivative. Fractional differential equations have gained considerable importance due to their varied applications in many fields of sciences and engineering. Recently, the numerical estimation of fractional differential equations has been discussed in the existing literature [24–31].
However, due to its implicit nature, very little theoretical results on the CCD method are known. Zhang [32] derived the truncation error representation of the CCD scheme when applied to 1D convection-diffusion equations and analyzed its oscillation property. Sengupta [11, 15] and Yu [12, 13] carefully studied the dispersion-relation of the CCD scheme, and proposed some improved CCD methods. To our best knowledge, the solvability and convergence of the CCD scheme for solving convection-diffusion equations have not been obtained in the existing literature.
The rest of the paper is organized as follows. Sect. 2 presents some lemmas and definitions for the proof. The proof of unique solvability is given in Sect. 3. And some conclusions are given in the final section.
Remark 1
2 Preliminaries
In this section, we will introduce some lemmas on circulant matrices which will be used to prove the main theorem in the next section.
Definition 1
([35])
Definition 2
([23])
Lemma 1
Proof
This lemma can be verified through direct computation. □
Lemma 2
([35])
For any two given circulant matrices A and B, the sum \(A+B\) is circulant, the product AB is circulant, and two matrices A, B commute, that is, \(AB=BA\).
Lemma 3
([36])
Remark 3
It is worth noting that if the matrix D is invertible, this lemma can be easily verified through matrix computation. In 2000, Silvester obtained the above results even if D is not invertible, which will be used in the proof of Theorem 1.
3 Unique solvability of CCD
For a positive integer M, let \(h = 1/M\). Discretize the interval \([0,1]\) into a uniform grid \(0=x_{0} < x_{1}<\cdots <x_{M-1}<x_{M}=1\), where \(x_{i}=ih\), \(i=0,\ldots,M\). Denote the numerical approximations of \(u(x_{i})\), \(u _{x}(x_{i})\), \(u_{xx}(x_{i})\) by \(U_{i}\), \(U_{i}^{\prime}\), \(U_{i}^{\prime\prime }\), respectively.
Theorem 1
Proof
4 Conclusions
A theoretical proof for the unique solvability of the CCD system for solving the 1D convection-diffusion equation with variable convection coefficients subject to the periodic boundary conditions is given in this paper.
This paper only focuses on the unique solvability of the CCD system for solving the convection-diffusion equation. Our future works will be focused on the convergence analysis of the CCD method and the generalization of the method for fractional order cases. And the CCD method for solving multi-dimensional elliptic boundary value problems [39–42] is also our future objective.
Declarations
Acknowledgements
Kejia Pan was supported by the National Natural Science Foundation of China (No. 41474103), the Excellent Youth Foundation of Hunan Province of China (No. 2018JJ1042), the Natural Science Foundation of Hunan Province of China (No. 2015JJ3148) and the Innovation-Driven Project of Central South University (No. 2018CX042). Hongling Hu was supported by the program for excellent talents in Hunan Normal University (No. ET1501). We are grateful to the three anonymous reviewers for their helpful comments.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992) MathSciNetView ArticleMATHGoogle Scholar
- Chu, P.C., Fan, C.: A three-point combined compact difference scheme. J. Comput. Phys. 140, 370–399 (1998) MathSciNetView ArticleMATHGoogle Scholar
- Chu, P.C., Fan, C.: A three-point sixth-order nonuniform combined compact difference scheme. J. Comput. Phys. 148, 663–674 (1999) MathSciNetView ArticleMATHGoogle Scholar
- Chu, P.C., Fan, C.: A three-point sixth-order staggered combined compact difference scheme. Math. Comput. Model. 32, 323–340 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Nihei, T., Ishii, K.: A fast solver of the shallow water equations on a sphere using a combined compact difference scheme. J. Comput. Phys. 187, 639–659 (2003) View ArticleMATHGoogle Scholar
- Chen, W., Chen, J.C., Lo, E.Y.: An interpolation based finite difference method on non-uniform grid for solving Navier–Stokes equations. Comput. Fluids 101, 273–290 (2014) MathSciNetView ArticleGoogle Scholar
- Ghader, S., Nordström, J.: High-order compact finite difference schemes for the vorticity-divergence representation of the spherical shallow water equations. Int. J. Numer. Methods Fluids 78, 709–738 (2015) MathSciNetView ArticleGoogle Scholar
- Cui, M.: Combined compact difference scheme for the time fractional convection-diffusion equation with variable coefficients. Appl. Math. Comput. 246, 464–473 (2014) MathSciNetMATHGoogle Scholar
- Takahashi, F.: Implementation of a high-order combined compact difference scheme in problems of thermally driven convection and dynamo in rotating spherical shells. Geophys. Astrophys. Fluid Dyn. 106, 231–249 (2012) MathSciNetView ArticleGoogle Scholar
- Gao, G.H., Sun, H.W.: Three-point combined compact difference schemes for time-fractional advection-diffusion equations with smooth solutions. J. Comput. Phys. 298, 520–538 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Sengupta, T.K., Lakshmanan, V., Vijay, V.: A new combined stable and dispersion relation preserving compact scheme for non-periodic problems. J. Comput. Phys. 228, 3048–3071 (2009) MathSciNetView ArticleMATHGoogle Scholar
- Yu, C.H., Wang, D., He, Z.T.: Pähtz. An optimized dispersion-relation-preserving combined compact difference scheme to solve advection equations. J. Comput. Phys. 300, 92–115 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Yu, C.H., Bhumkar, Y.G., Sheu, T.W.H.: Dispersion relation preserving combined compact difference schemes for flow problems. J. Sci. Comput. 62, 482–516 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Yu, C.H., Sheu, T.W.H.: Development of a combined compact difference scheme to simulate soliton collision in a shallow water equation. Commun. Comput. Phys. 19, 603–631 (2016) MathSciNetView ArticleMATHGoogle Scholar
- Sengupta, T.K., Vijay, V., Bhaumik, S.: Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties. J. Comput. Phys. 228, 6150–6168 (2009) View ArticleMATHGoogle Scholar
- Sun, H.W., Li, L.Z.: A CCD-ADI method for unsteady convection-diffusion equations. Comput. Phys. Commun. 185, 790–797 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Li, L.Z., Sun, H.W., Tam, S.C.: A spatial sixth-order alternating direction implicit method for two-dimensional cubic nonlinear Schrödinger equations. Comput. Phys. Commun. 187, 38–48 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Gao, G.H., Sun, H.W.: Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection-diffusion equations. Commun. Comput. Phys. 17, 487–509 (2015) MathSciNetView ArticleMATHGoogle Scholar
- He, D.D.: An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation. Numer. Algorithms 72, 1103–1117 (2016) MathSciNetView ArticleMATHGoogle Scholar
- He, D.D., Pan, K.J.: An unconditionally stable linearized CCD-ADI method for generalized nonlinear Schrodinger equations with variable coefficients in two and three dimensions. Comput. Math. Appl. 73, 2360–2374 (2017) MathSciNetView ArticleMATHGoogle Scholar
- He, D.D., Pan, K.J.: A fifth-order combined compact difference scheme for the Stokes flow on polar geometries. East Asian J. Appl. Math. 7, 714–727 (2018) MathSciNetView ArticleMATHGoogle Scholar
- Chen, B.Y., He, D.D., Pan, K.J.: A linearized high-order combined compact difference scheme for multi-dimensional coupled Burgers’ equations. Numer. Math., Theory Methods Appl. 11, 299–320 (2018) Google Scholar
- Lee, S.T., Liu, J., Sun, H.W.: Combined compact difference scheme for linear second-order partial differential equations with mixed derivative. J. Comput. Appl. Math. 264, 23–37 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Kumar, D., Singh, J., Baleanu, D.: A new numerical algorithm for fractional Fitzhugh-Nagumo equation arising in transmission of nerve impulses. Nonlinear Dyn. 91, 307–317 (2018) MathSciNetView ArticleGoogle Scholar
- Kumar, D., Agarwal, R.P., Singh, J.: A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. J. Comput. Appl. Math. 339, 405–413 (2018) MathSciNetView ArticleGoogle Scholar
- Singh, H., Srivastava, H.M., Kumar, D.: A reliable numerical algorithm for the fractional vibration equation. Chaos Solitons Fractals 103, 131–138 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Singh, J., Kumar, D., Qurashi, M.A., et al.: A novel numerical approach for a nonlinear fractional dynamical model of interpersonal and romantic relationships. Entropy 19, 375 (2017) View ArticleGoogle Scholar
- Yang, X.J., Gao, F., Srivastava, H.M.: A new computational approach for solving nonlinear local fractional PDEs. J. Comput. Appl. Math. 339, 285–296 (2018) MathSciNetView ArticleGoogle Scholar
- Yang, X.J., Gao, F., Srivastava, H.M.: Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations. Comput. Math. Appl. 73, 203–210 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Yang, X.J., Cao, F.: A new technology for solving diffusion and heat equations. Therm. Sci. 21, 133–140 (2017) View ArticleGoogle Scholar
- He, D.D., Pan, K.J.: An unconditionally stable linearized difference scheme for the fractional Ginzburg-Landau equation. Numer. Algorithms (2018). https://doi.org/10.1007/s11075-017-0466-y Google Scholar
- Zhang, J., Zhao, J.J.: Truncation error and oscillation property of the combined compact difference scheme. Appl. Math. Comput. 161, 241–251 (2005) MathSciNetMATHGoogle Scholar
- Hirsh, R.S.: Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique. J. Comput. Phys. 19, 90–109 (1975) MathSciNetView ArticleMATHGoogle Scholar
- Tian, Z.F.: A rational high-order compact ADI method for unsteady convection-diffusion equations. Comput. Phys. Commun. 182, 649–662 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Davis, P.J.: Circulant Matrices. Am. Math. Soc., Providence (2012) MATHGoogle Scholar
- Silvester, J.R.: Determinants of block matrices. Math. Gaz. 84, 460–467 (2000) View ArticleGoogle Scholar
- Yang, X.J., Baleanu, D., Khan, Y., et al.: Local fractional variational iteration method for diffusion and wave equations on Cantor sets. Rom. J. Phys. 59, 36–48 (2014) MathSciNetGoogle Scholar
- Bhrawy, A.H., Baleanu, D.: A spectral Legendre–Gauss–Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients. Rep. Math. Phys. 72, 219–233 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Pan, K.J., He, D.D., Hu, H.L., Ren, Z.Y.: A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems. J. Comput. Phys. 344, 499–515 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Pan, K.J., He, D.D., Hu, H.L.: An extrapolation cascadic multigrid method combined with a fourth-order compact scheme for 3D Poisson equation. J. Sci. Comput. 70, 1180–1203 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Pan, K.J., He, D.D., Chen, C.M.: An extrapolation cascadic multigrid method for elliptic problems on reentrant domains. Adv. Appl. Math. Mech. 9, 1347–1363 (2017) MathSciNetView ArticleGoogle Scholar
- Hu, H.L., Ren, Z.Y., et al.: On the convergence of an extrapolation cascadic multigrid method for elliptic problems. Comput. Math. Appl. 74, 759–771 (2017) MathSciNetView ArticleMATHGoogle Scholar