Abstract
This paper describes a method for convection-dominated fluid or heat flows. This method relies on the Hopmoc method and backward differentiation formulas. The present study discusses the convergence of the method when applied to a convection-diffusion equation in 1-D. The convergence analysis conducted produced sufficient conditions for the consistency analysis and used the von Neumann analysis to demonstrate that the method is stable. Moreover, the numerical results confirmed the conducted convergence analysis.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Oliveira, S., Kischinhevsky, M., Gonzaga de Oliveira, S.L.: Convergence analysis of the Hopmoc method. Int. J. Comput. Math. 86(8), 1375–1393 (2009)
Cabral, F., Osthoff, C., Costa, G., Brandão, D.N., Kischinhevsky, M., Gonzaga de Oliveira, S.L.: Tuning up TVD Hopmoc method on Intel MIC Xeon Phi architectures with Intel Parallel Studio Tools. In: Proceedings of the International Symposium on Computer Architecture and High Performance Computing Workshops (SBACPADW), Campinas, São Paulo, 17–20 October 2017, pp. 19–24 (2017)
Gordon, P.: Nonsymmetric difference equations. SIAM J. Appl. Math. 13, 667–673 (1965)
Gourlay, P.: Hopscotch: a fast second order partial differential equation solver. IMA J. Appl. Math. 6(4), 375–390 (1970)
Douglas, J.J., Russell, T.F.: Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal. 19(5), 871–885 (1955)
Robaina, D., Gonzaga de Oliveira, S., Kischinhevsky, M., Osthoff, C., Sena, A.: Numerical simulations of the 1-D modified Burgers equation. In: Proceedings of the 2019 Winter Simulation Conference (WSC), National Harbor, EUA, pp. 3231–3242 (2019)
Zhang, Z.: The multistep finite difference fractional steps method for a class of viscous wave equations. Math. Methods Appl. Sci. 34, 442–454 (2011)
Ding, H.F., Zhang, Y.X.: A new difference scheme with high accuracy and absolute stability for solving convection diffusion equations. J. Comput. Appl. Math. 230(2), 600–606 (2009)
Abbott, M.B.: An Introduction to the Method of Characteristics. Thames and Hudson Ltda, London (1996)
Cabral, F., Osthoff, C., Kischinhevsky, M., Brandão, D.: Hybrid MPI/OpenMP/OpenACC implementations for the solution of convection diffusion equations with Hopmoc method. In: Apduhan, B., Rocha, A.M., Misra, S., Taniar, D., Gervasi, O., Murgante, B., (eds.) 14th International Conference on Computational Science and its Applications (ICCSA), CPS, pp. 196–199. IEEE (2014)
Charney, J.G., Fjortoft, R., von Neumann, J.: Numerical integration of barotropic vorticity equation. Tellus 2, 237–254 (1950)
Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Robaina, D.T., de Oliveira, S.L.G., Kischinhevsky, M., Osthoff, C., Sena, A. (2020). A Convergence Analysis of a Multistep Method Applied to an Advection-Diffusion Equation in 1-D. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2020. ICCSA 2020. Lecture Notes in Computer Science(), vol 12249. Springer, Cham. https://doi.org/10.1007/978-3-030-58799-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-58799-4_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-58798-7
Online ISBN: 978-3-030-58799-4
eBook Packages: Computer ScienceComputer Science (R0)