Computational and Simulation Studies of the Laplacian Operator Containing 13-Points Stencils without Mixed Derivative in Cylindrical Mesh System
DOI:
https://doi.org/10.32350/sir.103.05Keywords:
cell dynamic simulation, cylindrical mesh system, discretization, explicit functions, finite difference method, Laplacian operator, 13-points stencilAbstract
Recently, considerable work has been done on graphic designing, artificial intelligence, optimization theory, solving numerical approximation of ordinary as well as partial differential equations by some numerical recipes. Discrete Laplacian operator is one of the leading tools used in various mathematical models and theories. It is the most demanding mathematical operator in Cell Dynamic Simulation (CDS), Self-Consistent Field Theory (SCFT) to study advanced materials, and image processing. Researchers are striving for a novel numerical scheme to acquire isotropic numerical results in addition to accuracy and stability. They are facing challenges in physically motivated discretization on curved surfaces/boundaries. In this work, a numerical study is carried out to discretize Laplacian operator on 13-point stencil in a cylindrical mesh system. The discretization process on Laplacian operator is carried out using the finite difference method, which allows fast computational competencies in simulation processes. This numerical study explores new formulations for the discretization of Partial Differential Equations (PDE) which are verified against analytic solutions and give consistent, convergent, accurate, and stable numerical results for 1st order and 2nd order partial derivatives and Laplacian operator.
Highlights
- A 13-point finite-difference stencil is developed for the cylindrical Laplacian operator.
- Taylor series expansion is used to discretize the Laplacian without mixed derivatives.
- Numerical simulations evaluate the accuracy, convergence, and stability of the proposed scheme.
Downloads
References
1. Mallah, R., Soomro, I., Ahmed, A., Muhammad, D., Latif, S. and Ali, I., 2023. Simulation of 13 points Laplacian operator in cylindrical mesh system by using explicit finite difference technique. VFAST Transactions on Mathematics, 11(1), pp.84-95.https://doi.org/10.21015/vtm.v11i1.1414
2. Mittal RC, Gahlaut S. High order finite difference schemes to solve Poisson's equation in cylindrical symmetry. Commun Appl Numer Methods. 1987;3(6):457-461. https://doi:10.1002/cnm.1630030603
3. Mallah, R., Siyal, W.A., Aslam, S., Sial, M.S. and Soomro, I., 2022. Using Crank-Nicolson’s scheme to discretize the Laplacian in a polar gird system with symmetric and asymmetric lines. VFAST Transactions on Mathematics, 10(2), pp.13-20.https://doi.org/10.21015/vtm.v10i2.1151
4. High order difference methods for heat equation in polar cylindrical coordinates. J Comput Phys. 1988;77(2):425-438. https://doi:10.1016/0021-9991(88)90176-3.
5. Soomro, I., Mallah, R., Iqbal, M.J., Ahmed, W. and Ghafoor, A., 2023. Discretization of laplacian operator on 19 points stencil using cylindrical mesh system with the help of explicit finite difference scheme. Pakistan Journal of Engineering, Technology and Science, 11(2), pp.1-10.https://doi.org/10.22555/pjets.v11i2.1019
6. Lai MC. A simple compact fourth-order Poisson solver on polar geometry. J Comput Phys. 2002;182(2):337-345.10.1006/jcph.2002.7172
7. Lai MC, Tseng CC. A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates. J Comput Appl Math. 2007;201(1):175-181. https://doi:10.1016/j.cam.2006.02.011.
8. Mohanty RK, Kumar R, Dahiya V. Cubic spline iterative method for Poisson's equation in cylindrical polar coordinates. ISRN Math Anal. 2012;2012:234516. https://doi:10.5402/2012/234516.
9. Tavakoli E, Lessani B, Hosseini R. High-order pole-treatment in cylindrical coordinates for incompressible flow simulations with finite-difference collocated schemes. J Comput Phys. 2015;300:571-590. https://doi:10.1016/j.jcp.2015.04.042
10. Bogey C, de Cacqueray N, Bailly C. Finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations. J Comput Phys. 2011;230(4):1134-1146. https://doi:10.1016/j.jcp.2010.10.031.
11. Tsega D, et al. Numerical solution of three-dimensional transient heat conduction equation in cylindrical coordinates. J Appl Math. 2022;2022:1993151.https:// doi:10.1155/2022/1993151.
12. Belkin, M., Sun, J., & Wang, Y. (2008, June). Discrete Laplace operator on meshed surfaces. In Proceedings of the twenty-fourth annual symposium on Computational geometry (pp. 278-287).https://doi.org/10.1145/1377676.137772
13. Malik, M. (2023). Lattice Green/Neumann function for the 2D Laplacian operator defined on square lattice on cylinders, tori and other geometries with some applications. Journal of Physics A: Mathematical and Theoretical, 56(21), 215202.https//:DOI 10.1088/1751-8121/accfd5
14. Zhang, Y., Shen, Y., Zhang, R., Liu, Y., Guo, Y., Deng, D. and Dinov, I.D., 2023. Numerical methods for computing the discrete and continuous Laplace transforms. arXiv preprint arXiv:2304.13204.https://doi.org/10.48550/arXiv.2304.13204
15. Huang, Y. and Oberman, A., 2014. Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM Journal on Numerical Analysis, 52(6), pp.3056-3084.https://doi.org/10.1137/140954040
16. Manea, D. (2026). Mesh-free numerical method for Dirichlet eigenpairs of the Laplacian with potential. arXiv preprint arXiv:2602.12008. https://doi.org/10.48550/arXiv.2602.12008
17. Abbasov, Z. D., Youssri, Y. H., Abdelfattah, W. M., Alraddadi, I., & Ahmad, H. (2026). Analytical and Numerical Solution of the Heat Conduction Equation in a Cylindrical Domain: Convergence Analysis and Bessel Collocation Method. International Journal of Analysis and Applications, 24, 237-237.https://doi.org/10.28924/2291-8639-24-2026-237
18. Huang, Y., & Oberman, A. (2016). Finite difference methods for fractional Laplacians. arXiv preprint arXiv:1611.00164.https://doi.org/10.48550/arXiv.1611.00164
19. Berntsson, F., & Eldén, L. (2001). Numerical solution of a Cauchy problem for the Laplace equation. Inverse Problems, 17(4), 839-853.https://DOI 10.1088/0266-5611/17/4/316
20. Maragakis, P., Soler, J., & Kaxiras, E. (2001). Variational finite-difference representation of the kinetic energy operator. Physical Review B, 64(19), 193101.https://doi.org/10.1103/PhysRevB.64.193101
21. Lai, M. C. (2001). A note on finite difference discretizations for Poisson equation on a disk. Numerical Methods for Partial Differential Equations: An International Journal, 17(3), 199-203.https://doi.org/10.1002/num.1Digital Object Identifier (DOI)
22. Morinishi, Y., Vasilyev, O. V., & Ogi, T. (2004). Fully conservative finite difference scheme in cylindrical coordinates for incompressible flow simulations. Journal of Computational Physics, 197(2), 686-710.10.1016/j.jcp.2003.12.015
23. Niakas, N., Loukopoulos, V. C., Douskos, C., Maroulis, G., & Simos, T. E. (2009). LFDM method on the Navier–Stokes equations in three-dimensional flow in cylindrical coordinates on the von Karman problem. AIP Conference Proceedings. https://doi.org/10.1063/1.3117136
24. Niakas, N., Loukopoulos, V. C., Douskos, C., Maroulis, G., & Simos, T. E. (2009). LFDM method on the Navier–Stokes equations in three-dimensional flow in cylindrical coordinates on the von Karman problem. AIP Conference Proceedings. [Companion publication of reference 1] https://doi.org/10.1063/1.3225372
25. Boyd, J. P., & Yu, F. (2010). Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, Logan–Shepp ridge polynomials, Chebyshev–Fourier series, cylindrical Robert functions, Bessel–Fourier expansions, square-to-disk conformal mapping and radial basis functions. Journal of Computational Physics. https://doi.org/10.1016/j.jcp.2010.11.011.
26. Volkov, V. M., Zherdetsky, A. A., Turovets, S. I., & Malony, A. D. (2010). A fast BiCG solver for the isotropic Poisson equation in the forward EIT problem in cylinder phantoms. Journal of Physics: Conference Series. https://doi.org/10.1088/1742-6596/224/1/012153.
27. Auteri, F., & Quartapelle, L. (2009). Spectral elliptic solvers in a finite cylinder. Communications in Computational Physics. [No DOI returned by the databases; journal page: https://www.global-sci.com/cicp/article/view/5612].
28. Golubovskiĭ, Yu. B., Timofeev, A. N., Gortschakow, S., Loffhagen, D., & Uhrlandt, D. (2009). Population of resonance and metastable atoms in a cylindrical volume of finite size. Physical Review E. https://doi.org/10.1103/PhysRevE.79.036409
29. Dalal, A., Eswaran, V., & Biswas, G. (2008). A finite-volume method for Navier–Stokes equations on unstructured meshes. Numerical Heat Transfer, Part B: Fundamentals. https://doi.org/10.1080/10407790802182653.
30. Chi, J., Liu, F., Xia, L., Shao, T., & Crozier, S. (2010). An improved cylindrical FDTD algorithm and its application to field-tissue interaction study in MRI. IEEE Transactions on Magnetics. https://doi.org/10.1109/TMAG.2010.2100098.
31. Chi, J., Liu, F., Xia, L., Shao, T., Mason, D. G., & Crozier, S. (2010). An improved cylindrical FDTD method and its application to field-tissue interaction study in MRI. IEEE Engineering in Medicine and Biology Society Annual International Conference. https://doi.org/10.1109/IEMBS.2010.5627189
32. Luginsland, T., Bühler, S., Obrist, D., & Kleiser, L. (2010). A parallel code for LES of compressible swirling jet flow undergoing vortex breakdown. PAMM – Proceedings in Applied Mathematics and Mechanics. https://doi.org/10.1002/pamm.201010348
33. Du, L. D., & Dalton, C. (2010). A CFD study of flow past a rotating cylinder to Re = 1000. Proceedings of the ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting (FEDSM-ICNMM2010). https://doi.org/10.1115/FEDSM-ICNMM2010-30100
34. Qian, Z., Zhang, J., & Li, C. (2010). Preconditioned pseudo-compressibility methods for incompressible Navier–Stokes equations. Science China Physics, Mechanics & Astronomy. https://doi.org/10.1007/s11433-010-4150-7
35. Juncu, G. (2007). A numerical study of momentum and forced convection heat transfer around two tandem circular cylinders at low Reynolds numbers. Part I: Momentum transfer. International Journal of Heat and Mass Transfer. https://doi.org/10.1016/j.ijheatmasstransfer.2007.02.020
36. da Silva, W. P., Silva, C. M. D. P. S., Silva, D. D. P. S., & Silva, C. D. P. S. (2008). Numerical simulation of the water diffusion in cylindrical solids. International Journal of Food Engineering. https://doi.org/10.2202/1556-3758.1394.
37. Sheu, Y.-L., & Li, P.-C. (2008). Simulations of photoacoustic wave propagation using a finite-difference time-domain method with Berenger's perfectly matched layers. Journal of the Acoustical Society of America. https://doi.org/10.1121/1.3003087.
38. Sheu, Y.-L., & Li, P.-C. (2009). Simulations of thermally induced photoacoustic wave propagation using a pseudospectral time-domain method. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control. https://doi.org/10.1109/TUFFC.2009.1144
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Rabnawaz Mallah, Abdul Rehman Soomro, Inayatullah Soomro, Fida Hussain, Nawab Khan
This work is licensed under a Creative Commons Attribution 4.0 International License.
