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The Numerov process over a non-uniform grid
Published version of a preprint
Abstract
The Numerov process is a solution method applicable to some classes of differential equations, that provides an error term of the fifth order in the grid size with a computational cost comparable to that of the finite-difference scheme. In the original formulation of the method, a uniform grid size is required; the paper shows a procedure for extending its applicability to a non-uniform grid in one dimension. The effectiveness of the procedure is tested on a model problem, and comparisons with other methods are carried out. Finally, it is shown how to extend the applicability of the method to a larger class of equations; among these, the mathematical model of semiconductor devices is important in view of its applications to the integrated-circuit technology.
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Cited by 2
Research works printed but not linked: 8
- Chow, P.C.: Computer solutions to the Schrödinger equation. Am. J. Phys. 40, 730–734 (1972) · DOI
- Vigo-Aguiar, J., Ramos, H.: A variable-step Numerov method for the numerical solution of the Schrödinger equation. J. Math. Chem. 37(3), 255–262 (2005) · DOI · Paper page
- Press, W.H., Flannery, B.R., Teukolsky, S.A., Wetterling, W.T.: Numerical recipes: the art of scientific computing. Cambridge University Press, New York (1988)
- Zlotnik, A.: $$\check{\rm C}$$iegis, R.: A compact higher-order finite-difference scheme for the wave equation can be strongly non-dissipative on non-uniform meshes. Appl. Math. Lett. 115, 106949 (2021) · DOI
- Jain, M.K., Iyengar, S.R.K., Subramanyam, G.S.: Variable mesh methods for the numerical solution of two-point singular perturbation problems. Comput. Methods Appl. Mech. Eng. 42, 273–286 (1984) · DOI
- Kierzenka, J., Shampine, L.F.: A BVP solver based on residual control and the MATLAB PSE. ACM Trans. Math. Softw. 27(3), 299–316 (2001) · DOI
- Speciale, N., Brunetti, R., Rudan, M.: Extending the Numerov process to the semiconductor transport equations. In: Esseni, D., Palestri, P., Rideau, D. (eds.) Proceedings of the 2019 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD 2019), pp. 37–40. IEEE, Udine, Italy (2019) · DOI
- Speciale, N., Brunetti, R., Rudan, M.: Solution of the semiconductor-device equations by the Numerov process. Adv. Sci. Technol. Eng. Syst. J. 5(6), 1414–1421 (2020) · DOI
Metadata
- Authors
- Brunetti, R., Speciale, N., & Rudan, M.
- Venue
- Journal of Computational Electronics
- Published
- 2021-04-13
- Volume
- 20
- Issue
- 3
- Pages
- 1105-1113
- ISSN
- 1569-8025
- 1572-8137
- Cited by
- 2
- Language
- en
- Licence
- creativecommons.org/licenses/by/4.0 (tdm)
- creativecommons.org/licenses/by/4.0 (vor)
- Funders
- Alma Mater Studiorum - Università di Bologna
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- Brunetti, R., Speciale, N., & Rudan, M. (2021). The Numerov process over a non-uniform grid. Journal of Computational Electronics. https://doi.org/10.1007/s10825-021-01699-3
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