Document Type : Original Article

**Authors**

Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), Iran

**Abstract**

In this paper, we construct a difference potentials method (DPM) based on the locally one-dimensional (LOD) technique to solve the two-dimensional nonlinear convection-diffusion interface problems. The advantage of using the LOD scheme is that the linear system resulting from the auxiliary problem has a simpler structure and can be solved efficiently and accurately with less central processor (CPU) time. Numerical results validate the robustness, accuracy, and efficiency of the proposed method.

**Keywords**

**Main Subjects**

[1] J. Albright, Y. Epshteyn, M. Medvinsky, and Q. Xia, High-order numerical schemes based on difference potentials for 2D elliptic problems with material interfaces, Appl. Numer. Math., 111 (2017), pp. 64–91.

[2] A. R. Appadu and H. H. Gidey, Time-splitting procedures for the numerical solution of the 2d advectiondiffusion equation, Mathematical Problems in Engineering, 2013 (2013), p. 634657.

[3] D. S. Britt, S. V. Tsynkov, and E. Turkel, A high-order numerical method for the Helmholtz equation with nonstandard boundary conditions, SIAM J. Sci. Comput., 35 (2013), pp. A2255–A2292.

[4] S. Britt, S. Petropavlovsky, S. Tsynkov, and E. Turkel, Computation of singular solutions to the Helmholtz equation with high order accuracy, Appl. Numer. Math., 93 (2015), pp. 215–241.

[5] Y. Epshteyn, Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model, J. Sci. Comput., 53 (2012), pp. 689–713.

[6] Y. Epshteyn and M. Medvinsky, On the solution of the elliptic interface problems by difference potentials method, in Spectral and high order methods for partial differential equations—ICOSAHOM 2014, vol. 106 of Lect. Notes Comput. Sci. Eng., Springer, Cham, 2015, pp. 197–205.

[7] S. Huang and Y. Liu, A fast multipole boundary element method for solving the thin plate bending problem, Eng. Anal. Bound. Elem., 37 (2013), pp. 967–976.

[8] E. Lee and D. Kim, Stability analysis of the implicit finite difference schemes for nonlinear Schr¨odinger equation, AIMS Math., 7 (2022), pp. 16349–16365.

[9] J. Liu and Z. Zheng, IIM-based ADI finite difference scheme for nonlinear convection-diffusion equations with interfaces, Appl. Math. Model., 37 (2013), pp. 1196–1207.

[10] M. Medvinsky, S. Tsynkov, and E. Turkel, The method of difference potentials for the Helmholtz equation using compact high order schemes, J. Sci. Comput., 53 (2012), pp. 150–193.

[11] , High order numerical simulation of the transmission and scattering of waves using the method of difference potentials, J. Comput. Phys., 243 (2013), pp. 305–322.

[12] , Solving the Helmholtz equation for general smooth geometry using simple grids, Wave Motion, 62 (2016), pp. 75–97.

[13] A. A. Reznik, Approximation of surface potentials of elliptic operators by difference potentials, Dokl. Akad. Nauk SSSR, 263 (1982), pp. 1318–1321.

[14] V. S. Ryaben’kii, Method of difference potentials and its applications, vol. 30 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 2002. Translated from the 2001 Russian original by Nikolai K. Kulman.

[15] V. S. Ryaben’kii, V. I. Turchaninov, and E. Y. Epshte ` ˘ın, An algorithm composition scheme for problems in composite domains based on the method of difference potentials, Zh. Vychisl. Mat. Mat. Fiz., 46 (2006), pp. 1853–1870.

[16] Q. Sheng, The legacy of ADI and LOD methods and an operator splitting algorithm for solving highly oscillatory wave problems, in Modern mathematical methods and high performance computing in science and technology, vol. 171 of Springer Proc. Math. Stat., Springer, Singapore, 2016, pp. 215–230.

[17] G. D. Smith, Numerical solution of partial differential equations, Oxford Applied Mathematics and Computing Science Series, The Clarendon Press, Oxford University Press, New York, third ed., 1985. Finite difference methods.

[18] D. A. Voss and A. Q. M. Khaliq, Parallel LOD methods for second order time dependent PDEs, Comput. Math. Appl., 30 (1995), pp. 25–35.

[19] R. F. Warming and B. J. Hyett, The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys., 14 (1974), pp. 159–179.

[20] H. Zhu, H. Shu, and M. Ding, Numerical solutions of two-dimensional Burgers’ equations by discrete Adomian decomposition method, Comput. Math. Appl., 60 (2010), pp. 840–848.