A monotonic weighted compact finite difference solution for a non-linear steady advection-diffusion equation

Chivapornthip, Prapol

doi:10.22034/cmde.2024.61722.2680

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	45
تعداد شماره‌ها	1,492
تعداد مقالات	18,196
تعداد مشاهده مقاله	58,885,901
تعداد دریافت فایل اصل مقاله	20,371,215

	A monotonic weighted compact finite difference solution for a non-linear steady advection-diffusion equation
Computational Methods for Differential Equations
مقاله 3، دوره 14، شماره 2، تیر 2026، صفحه 549-566 اصل مقاله (1.41 M)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.61722.2680
نویسنده
Prapol Chivapornthip^*
Department of Industrial Engineering, Faculty of Engineering, Kasetsart University, Thailand.
چکیده
This paper introduces a monotonic weighted compact finite difference scheme (WC-FDM) designed to solve the nonlinear one-dimensional steady advection-diffusion equation (ADE). The WC-FDM scheme is validated against the analytical solution and is adaptable to accommodate both uniform and non-uniform grid spacing. Criteria for selecting weights have been developed to ensure scheme monotonicity. Computational performance is benchmarked against other numerical schemes. Numerical analyses reveal that the WC-FDM accurately solves the non-linear steady ADE for both uniform and non-uniform grid spacing scenarios without introducing spurious oscillations. The proposed weight criteria maintain the monotonicity of the WC-FDM scheme resulting the computational stability regardless of the advection dominance level and grid spacing uniformity.
کلیدواژه‌ها
Compact finite difference؛ Monotonic scheme؛ Advection-diffusion؛ Weighted finite difference

مراجع
[1] M. Abdullah, M. Yaseen, and M. D. L. Sen, An efficient collocation method based on Hermite formula and cubic B-splines for numerical solution of the Burgers’ equation, Math. Comput. Simul., 197 (2022), 166–184. [2] J. D. Anderson, Computational Fluid Dynamics: The Basics with Applications, McGraw-Hill, 1995. [3] J. B. Angel, J. W. Banks, A. M. Carson, and W. D. Henshaw, Efficient Upwind Schemes for Linear and Nonlinear Dispersive Maxwell’s Equations on Overset Grids, 2023 International Applied Computational Electromagnetics Society Symposium (ACES), (2023), 1–2. [4] J. W. Banks and W. D. Henshaw, Upwind schemes for the wave equation in second-order form, J. Comput. Phys., 231(17) (2012), 5854–5889. [5] S. F. Bradford and N. D. Katopodes, The anti-dissipative, non-monotone behavior of Petrov–Galerkin upwinding, Int. J. Numer. Methods Fluids, 33 (2000), 583–608. [6] H. Charles, Numerical Computation of Internal and External Flows, Butterworth-Heinemann, 2007. [7] P. Chivapornthip, A Singular Perturbation Solution of Power Law Fluid Flow in Open Channel Tube by Using Quadratic B-spline Collocation Method, J. Appl. Eng. Sci., 24(1) (2020), 33 – 41. [8] L. Correa, G. A. B. Lima, M. A. C. Candezano, M. P. S. Braun, C. M. Oishi, H. A. Navarro, and V. G. Ferreira, A C2-continuous high-resolution upwind convection scheme, Int. J. Numer. Methods Fluids, 72 (2013), 1263–1285. [9] M. Dehghan, Weighted finite difference techniques for the one-dimensional advection–diffusion equation, Appl. Math. Comput., 147(2) (2004), 307–319. [10] U. Erdogan, Improved upwind discretization of the advection equation, Numer. Methods Partial Differ. Equ., 30 (2014), 773-787. [11] M. E. Fiadeiro and G. Veronis, On weighted-mean schemes for the finite-difference approximation to the advectiondiffusion equation, Tellus, 29 (1977), 512–522. [12] H. Karahan, Solution of Weighted Finite Difference Techniques With the Advection Diffusion Equation Using Spreadsheets, Comput. Appl. Eng. Educ., 16 (2005), 147–156. [13] G. V. Krivovichev and S. A. Mikheev, Stability analysis of schemes with upwind differences for the solution of the system of kinetic equations for the modelling of semi-compressible gas, International Conference “Stability and Control Processes” in Memory of V.I. Zubov (SCP), (2015), 399–401. [14] S. Kutluay, A. Esen, and I. Dag, Numerical solutions of the Burgers’ equation by the least-squares quadratic B-spline finite element method, J. Comput. Appl. Math., 167(1) (2004), 21–33. [15] P. W. Li and F. Zhang, A weighted–upwind generalized finite difference (WU–GFD) scheme with high–order accuracy for solving convection–dominated problems, Appl. Math. Lett., 150 (2024), 1–7. [16] W. Liao, An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation, Appl. Math. Comput., 206(2) (2008), 755–764. [17] W. Liao, A fourth-order finite-difference method for solving the system of two-dimensional Burgers’ equations, Int. J. Numer. Methods Fluids, 64 (2010), 565–590. [18] P. R. M. Lyra and K. Morgan, A review and comparative study of upwind biased schemes for compressible flow computation Part I: 1D first-order schemes, Arch. Comput. Methods Eng., 7 (2000), 19–55. [19] A. E. Mattsson and W. J. Rider, Artificial viscosity: back to the basics, Int. J. Numer. Methods Fluids, 77 (2015), 400–417. [20] R. C. Mittal and R. K. Jain, Numerical solutions of nonlinear Burgers’ equation with modified cubic B-splines collocation method, Appl. Math. Comput., 218(15) (2012), 7839–7855. [21] T. Oziş and U. Erdoğkan, An exponentially fitted method for solving Burgers’ equation, Int. J. Numer. Methods Eng., 79 (2009), 696–705. [22] M. A. Ramadan, T. S. El-Danaf, and F. E. I. A. Alaal, A numerical solution of the Burgers’ equation using septic B-splines, Chaos Solit. Fractals, 26(4) (2005), 1249–1258. [23] B. Saka and I. Dağ, Quartic B-spline collocation method to the numerical solutions of the Burgers’ equation, Chaos Solit. Fractals, 32(3) (2007), 1125–1137. [24] Y. V. S. S. Sanyasiraju and G. Chandhini, A note on two upwind strategies for RBF-based grid-free schemes to solve steady convection–diffusion equations, Int. J. Numer. Methods Fluids, 61 (2009), 1053–1062. [25] M. Sari and G. Gürarslan, A sixth-order compact finite difference scheme to the numerical solutions of Burgers’ equation, Appl. Math. Comput., 208(2) (2009), 475–483. [26] A. Shaw and D. Roy, Stabilized SPH-based simulations of impact dynamics using acceleration-corrected artificial viscosity, Int. J. Impact Eng., 48 (2012), 98–106. [27] M. Tamsir, N. Dhiman, and V. K. Srivastava, Extended modified cubic B-spline algorithm for nonlinear Burgers’ equation, Beni-Suef Univ. J. Basic Appl. Sci., 5(3) (2016), 244–254. [28] B. J. R. Thornber and D. Drikakis, Numerical dissipation of upwind schemes in low Mach flow, Int. J. Numer. Methods Fluids, 56 (2008), 1535–1541. [29] F. Xiao, X. Tang, and H. Ma, High-order US-FDTD based on the weighted finite-difference method, Microw. Opt. Technol. Lett., 45 (2005), 142–144. [30] S. Xiel, G. Li, S. Yi, and S. Heo, A compact finite difference method for solving Burgers’ equation, Int. J. Numer. Methods Fluids, 62 (2010), 747–764. [31] D. Xiu and G. E. Karniadakis, Supersensitivity due to uncertain boundary conditions, Int. J. Numer. Methods Eng., 61 (2004), 2114–2138. [32] X. Yang, Y. Ge, and B. Lan, A class of compact finite difference schemes for solving the 2D and 3D Burgers’ equations, Math. Comput. Simul., 185 (2021), 510–534. [33] X. Yang, Y. Ge, and L. Zhang, A class of high-order compact difference schemes for solving the Burgers’ equations, Appl. Math. Comput., 358 (2019), 394–417. [34] Y. Yang, M. Li, S. Shu, and A. Xiao, High order schemes based on upwind schemes with modified coefficients, J. Comput. Appl. Math., 195 (2006), 242–251. [35] R. Yeh, Y. S. G. Nashed, T. Peterka, and X. Tricoche, Fast Automatic Knot Placement Method for Accurate B-spline Curve Fitting, Comput. Aided Des., 128 (2020), 1–13. [36] P. G. Zhang and J. P. Wang, A predictor–corrector compact finite difference scheme for Burgers’ equation, Appl. Math. Comput., 219 (2012), 892–898.
آمار تعداد مشاهده مقاله: 209 تعداد دریافت فایل اصل مقاله: 310

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

A monotonic weighted compact finite difference solution for a non-linear steady advection-diffusion equation