A high order convergent adaptive numerical method for singularly perturbed nonlinear systems

Springer Science and Business Media LLC - Tập 41 - Trang 1-16 - 2022
Sumit1, Shashikant Kumar1, Sunil Kumar1
1Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, India

Tóm tắt

In this work, we develop a high order convergent adaptive numerical method for a system of first-order singularly perturbed nonlinear differential equations with distinct perturbation parameters. The problem is discretized by a hybrid finite difference scheme for which a posteriori error estimate in the maximum norm is derived. The layer-adapted meshes are generated using equidistribution of the monitor function chosen based on the derived a posteriori error estimate. Numerical results are presented that validate the theory and show the effectiveness of the present numerical method.

Tài liệu tham khảo

Amiraliyev G (2005) The convergence of a finite difference method on layer-adapted mesh for a singularly perturbed system. Appl Math Comput 162(3):1023–1034 Antoulas AC (2005) Approximation of large-scale dynamical systems. SIAM, USA Ascher UM, Mattheij RMM, Russell RD (1994) Numerical solution of boundary value problems for ordinary differential equations, vol 13. SIAM, USA Atkinson KE (2008) An introduction to numerical analysis. Wiley, New York Boor CD (1973) Good approximation by splines with variable knots, in: spline functions and approximation theory. In: Proceedings of the symposium held at the University of Alberta, Edmonton, Birkhauser, Basel Cen Z, Xu A, Le A (2010) A second-order hybrid finite difference scheme for a system of singularly perturbed initial value problems. J Comput Appl Math 234(12):3445–3457 Chang KW, Howes FA (1984) Nonlinear singular perturbation phenomena. Springer, New York Das P (2015) Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems. J Comput Appl Math 290:16–25 Gajic Z (2001) Optimal control of singularly perturbed linear systems and applications. CRC Press, London Gupta V, Sahoo SK, Dubey RK (2021) Robust higher order finite difference scheme for singularly perturbed turning point problem with two outflow boundary layers. Comput Appl Math 40:179 Huang J, Cen Z, Xu A, Liu L-B (2020) A posteriori error estimation for a singularly perturbed volterra integro-differential equation. Numer Algorithms 83(2):549–563 Huang J, Cen Z, Xu A (2021) An improved a posteriori error estimation for a parameterized singular perturbation problem. Appl Math Lett 114:106912 Kadalbajoo MK, Gupta V (2010) A brief survey on numerical methods for solving singularly perturbed problems. Appl Math Comput 217(8):3641–3716 Kadalbajoo MK, Patidar KC (2003) Singularly perturbed problems in partial differential equations: a survey. Appl Math Comput 134:371–429 Kopteva N, Stynes M (2001) A robust adaptive method for a quasi-linear one-dimensional convection–diffusion problem. SIAM J Numer Anal 39(4):1446–1467 Kumar S, Kumar M (2012) Parameter-robust numerical method for a system of singularly perturbed initial value problems. Numer Algorithms 59(2):185–195 Kumar S, Kumar M (2016) Analysis of some numerical methods on layer adapted meshes for singularly perturbed quasilinear systems. Numer Algorithms 71(1):139–150 Kumar S, Kumar S, Sumit (2020) High-order convergent methods for singularly perturbed quasilinear problems with integral boundary conditions. Math Methods Appl Sci. https://doi.org/10.1002/mma.6854 Kumar S, Kumar S, Sumit (2021) A posteriori error estimation for quasilinear singularly perturbed problems with integral boundary condition. Numer Algorithms. https://doi.org/10.1007/s11075-021-01134-5 Ladde GS, Lakshmikantham V, Vatsala AS (1985) Monotone iterative techniques for nonlinear differential equations, vol 27. Pitman Publishing, London Liu L-B, Long G, Cen Z (2020) A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition. Numer Algorithms 83(2):719–739 Meenakshi PM, Valarmathi S, Miller JJH (2010) Solving a partially singularly perturbed initial value problem on shishkin meshes. Appl Math Comput 215(9):3170–3180 Munyakazi JB, Patidar KC, Sayi MT (2019) A fitted numerical method for parabolic turning point singularly perturbed problems with an interior layer. Numer Methods Partial Differ Equ 35:2407–2422 Podila PC, Kumar K (2020) A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs. Comput Appl Math 39:140 Raj I, Johnson PM, Miller JJH, Sigamani V (2016) A parameter uniform almost first order convergent numerical method for non-linear system of singularly perturbed differential equations. Biomathematics 5(2):Article ID: 1608111 Ramos H, Vigo-Aguiar J (2008) A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems. Int J Comput Math 85(3–4):603–611 Rao SCS, Kumar S (2012) Second order global uniformly convergent numerical method for a coupled system of singularly perturbed initial value problems. Appl Math Comput 219(8):3740–3753 Roos H-G, Stynes M, Tobiska L (2008) Robust numerical methods for singularly perturbed differential equations: convection-diffusion-reaction and flow problems, vol 24. Springer, Berlin Saksena V, O’Reilly J, Kokotovic P (1984) Singular perturbations and time-scale methods in control theory: survey 1976–1983. Automatica 20(3):273–293 Sumit, Kumar S, Vigo-Aguiar J (2021) Analysis of a nonlinear singularly perturbed volterra integro-differential equation. J Comput Appl Math. https://doi.org/10.1016/j.cam.2021.113410 Varga RS (1962) Iterative analysis. Springer, Berlin Xu X, Mathur R, Jiang J, Rogers G, Kundur P (1998) Modeling of generators and their controls in power system simulations using singular perturbations. IEEE Trans Power Syst 13(1):109–114 Xu X, Huang W, Russell RD, Williams JF (2011) Convergence of de boor’s algorithm for the generation of equidistributing meshes. IMA J Numer Anal 31(2):580–596 Zhang Y, Naidu DS, Cai C, Zou Y (2014) Singular perturbations and time scales in control theories and applications: an overview 2002–2012. Int J Inf Syst Sci 9(1):1–36