Abstract
In the context of finite difference approximations and semi-discretization methods, empirical criteria of adaptive gridding, based on the concentration of the nodes in the regions of high spatial derivatives and motion of them at a prescribed velocity, are generally used. The purpose of the present paper is to give a contribution to overcome the gap between the practical use of those criteria and its theoretical justification. Finite difference discretizations of transport equations and convection-diffusion equations are considered. It is proved that if the mesh density is proportional to the spatial gradient and the nodes are moved at the convection speed, then the spatial truncation error is minimized.
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References
H. A. Dwyer and B. R. Sanders, Numerical modelling of unsteady flame propagation, Acta Astronaut. 5 (1978), 1171–1184.
G. Eigenberger, Modelling and computer simulation as aids for safe reactor operation, Ger. Chem. Eng. 3 (1980), 211–216.
H. Guillard and R. Peyret, On the use of spectral methods for the numerical solution of stiff problems, Comp. Meth. Appl. Mech. and Eng. 66 (1988), 17–43.
B. Larouturou, Adaptive numerical methods for unsteady flame propagation, in Proceedings of the 1985 AMS-SIAM Summer Seminar on Reacting flows: Combustion and Chemical Reactors, eds. G. S. S. Ludford, (to appear).
G.J. McRae, W. R. Goodin and J. H. Seinfeld, Numerical solution of the atmospheric diffusion equation for chemically reacting flows, J. Comput. Phys. 45 (1982), 1–42.
K. Miller and R. N. Miller, Moving finite elements I, SIAM J. Numer. Anal. 18 (6) (1981), 1019–1032.
K. Miller, Moving finite elements II, SIAM J. Numer. Anal. 18 (6) (1981), 1033–1057.
H. S. Price, R. S. Varga and J. E. Warren, Application of oscillation matrices to convection-diffusion equations, J. Math. Phys. 45 (1966), 301–311.
J.I. Ramos, The application of finite difference and finite element methods to a reaction-diffusion system in combustion, in Numerical Methods in Laminar and Turbulent Flow, eds. C. Taylor, J. A. Johnson and W. R Smith, Pineridge Press, Swansea, U. K. (1983), 1137–1147.
R. Varga, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs (1962).
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© 1988 Springer Basel AG
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de Oliveira, P., Oliveira, F.A. (1988). On a Theoretical Justification of Adaptive Gridding for Finite Difference Approximations. In: Agarwal, R.P., Chow, Y.M., Wilson, S.J. (eds) Numerical Mathematics Singapore 1988. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série internationale d’Analyse numérique, vol 86. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6303-2_32
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DOI: https://doi.org/10.1007/978-3-0348-6303-2_32
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-7643-2255-7
Online ISBN: 978-3-0348-6303-2
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