Abstract
In this paper, we study the stability of the Crank–Nicolson and Euler schemes for time-dependent diffusion coefficient equations on a staggered grid with explicit and implicit approximations to the Dirichlet boundary conditions. Using the matrix representation for the numerical scheme and boundary conditions it is shown that for implicit boundary conditions the Crank–Nicolson scheme is unrestrictedly stable while it becomes conditionally stable for explicit boundary conditions. Numerical examples are provided illustrating this behavior. For the Euler schemes the results are similar to those for the constant coefficient case. The implicit Euler with implicit or explicit boundary conditions is unrestrictedly stable while the explicit Euler with explicit boundary conditions presents the usual stability restriction on the time step.
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References
Axelson, O.: Error estimates over infinite intervals of some discretizations of evolution equations. BIT 24, 413–424 (1984)
Barnes, H.A.: Thixotropy—a review. J. Non-Newton Fluid 70, 1–33 (1997)
Chua, T.S., Dew, P.M.: The design of a variable-step integrator for the simulation of gas transmisson network. Int. J. Numer. Meth. Eng. 20, 1797–1813 (1984)
Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1975)
Danskin, J.M.: The theory of max–min with applications. SIAM J. Appl. Math. 14, 641–664 (1966)
Guler, O.: Foundations of Optimization in Finite Dimensions. Springer-Verlag, Berlin (2010)
Huang, K.M., Lin, Z., Yang, X.Q.: Numerical simulation of microwave heating on chemical reaction in dilute solution. Prog. Eletromagn. Res. 49, 273–289 (2004)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1996)
Hunsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, Berlin (2003)
McCartin, B.J., Labadie, S.M.: Accurate and efficient pricing of vanilla stock options via the Crandall–Douglas scheme. Appl. Math. Comput. 143, 39–60 (2003)
Oishi, C.M., Cuminato, J.A., Yuan, J.Y., McKee, S.: Stability of numerical schemes on staggered grids. Numer. Linear. Algebr. 15, 945–967 (2008)
Pozar, D.M.: Microwave Engineering. John-Wiley, New york (2005)
Sousa, E.: On the edge of stability analysis. Appl. Numer. Math. 59, 1322–1336 (2009)
Sucec, J.: Practical stability analysis of finite difference equations by the matrix method. Int. J. Numer. Meth. Eng. 24, 679–687 (1987)
Tadjeran, C.: Stability analysis of the Crank–Nicholson method for variable coefficient diffusion equation. Commun. Numer. Meth. En. 23, 29–34 (2007)
Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. Princeton University Press, New Jersey (2005)
Wilmott, P., Howison, S., Dewynne, J.: The Mathematics of Financial Derivatives: A Student Introduction. Cambridge University Press, Cambridge (1995)
Yueh, W.C.: Eigenvalues of several tridiagonal matrices. Appl. Math. E-Notes 5, 66–74 (2005)
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We gratefully acknowledge the financial support given by FAPESP(Fundação de Amparo a Pesquisa do Estado de São Paulo), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico).
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Communicated by Anna-Karin Tornberg.
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Oishi, C.M., Yuan, J.Y., Cuminato, J.A. et al. Stability analysis of Crank–Nicolson and Euler schemes for time-dependent diffusion equations. Bit Numer Math 55, 487–513 (2015). https://doi.org/10.1007/s10543-014-0509-x
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DOI: https://doi.org/10.1007/s10543-014-0509-x
Keywords
- Stability analysis
- Crank–Nicolson scheme
- Staggered grids
- Boundary conditions
- Non-constant coefficient diffusion equations