Using Crank-Nicolson’s scheme to discretize the Laplacian in a polar gird system with symmetric and asymmetric lines


  • Rabnawaz Mallah Department of Mathematics, Shah Abdul Latif University, KhairpurMirsDistrict-66020,Pakistan,
  • Wajid Ahmed Siyal
  • Saira Aslam University Malaysia Sarawak
  • Muhammad Suleman Sial Department of Basic Sciences and Related Studies,Mehran University of Engineering and Technology, Pakistan
  • Inayatullah Soomro Shah Abdul Latif University, Khairpur, Pakistan



Numerous techniques exist for solving and describing the Partial differential equation’s mathematical and computational model. The Laplacian operator is one of the most effective techniques for solving linear and nonlinear partial differential equations. It is quick, and researchers use it frequently because of its modern technique and high accuracy in results. The Crank-Nicolson (CN) scheme in the Cartesian coordinate system has been discussed in this research work. Using this method, a numerical approximation scheme in Cartesian coordinate
system has been discretized on a 5 point stencil, extendable to nine points. The Tailor Series was used to discretize this scheme on 5-point stencils, which will be used in FORTRAN code for numerical approximation and can be visualized in OPEDX software. The Nicolson scheme is a finite difference scheme used to solve partial differential equations such as heat, wave, and diffusion equations in both 1-D and 2-D. Because of his extendable stencil, it will create accuracy and stability in the novel results of the scheme. These extendable stencils will reduce the error of the scheme and will assist researchers in finding novel results by solving ODES and PDES using the CN method.




Antoine, X. and Besse, C. [2003], ‘Unconditionally stable discretization schemes of non-reflecting boundary conditions for the one-dimensional schrödinger equation’, Journal of Computational Physics 188(1), 157–175.

Bruno-Alfonso, A., Cabezas-Gómez, L., Navarro, H. A. et al. [2012], ‘Alternate treatments of jacobian

singularities in polar coordinates within finite-difference schemes’, World Journal of Modelling and Simulation

(3), 163–171.

Campin, J.-M., Adcroft, A., Hill, C. and Marshall, J. [2004], ‘Conservation of properties in a free-surface

model’, Ocean Modelling 6(3-4), 221–244.

Çelik, C. and Duman, M. [2012], ‘Crank–nicolson method for the fractional diffusion equation with the riesz fractional derivative’, Journal of computational physics 231(4), 1743–1750.

Desbrun, M., Kanso, E. and Tong, Y. [2006], Discrete differential forms for computational modeling, in ‘ACM SIGGRAPH 2006 Courses’, pp. 39–54.

Esmaeilzadeh, M. [2016], A Cartesian Cut-Stencil Method for the Finite Difference Solution of PDEs in Complex Domains, PhD thesis, University of Windsor (Canada).

Hamilton, B. and Bilbao, S. [2013], Fourth-order and optimised finite difference schemes for the 2-d wave equation, in ‘Proc. 16th Conference on Digital Audio Effects (DAFx-13)’.

Huiskamp, G. [(1991)], ‘Difference formulas for the surface laplacian on a triangulated surface’, Journal of computational physics 95(2), 477–496.

Latif, S., Mallah, R. and Soomro, I. [2021], ‘Discretization of laplacian operator in polar coordinates system on 9-point stencil with mixed pde’s derivative approximation using finite difference method’, Journal of Mathematical Sciences & Computational Mathematics 2(3), 387–394.

Leclaire, S., El-Hachem, M., Trépanier, J.-Y. and Reggio, M. [2014], ‘High order spatial generalization of 2d and 3d isotropic discrete gradient operators with fast evaluation on gpus’, Journal of Scientific Computing 59(3), 545–573.

LeVeque, R. J. [2007], Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems, SIAM.

Mallah, R. and Soomro, I. [2022], ‘Comparative study of numerical approximation schemes for laplacian operator in polar mesh system on 9-points stencil including mixed partial derivative by finite difference method’, Journal of Mathematical Sciences & Computational Mathematics 3(4), 516–525.

Ramadugu, R., Thampi, S. P., Adhikari, R., Succi, S. and Ansumali, S. [2013], ‘Lattice differential operators for computational physics’, EPL (Europhysics Letters) 101(5), 50006.

Reimer, A. S. and Cheviakov, A. F. [2013], ‘A matlab-based finite difference solver for the poisson problem with mixed dirichlet neumann boundary conditions’, Computer Physics Communications 184(3), 783–798.




How to Cite

Mallah, R., Siyal, W. A., Aslam, S., Sial, M. S., & 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), 13–20.