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Journal of Scientific Computing

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01-05-2015

Semi-analytical Time Differencing Methods for Stiff Problems

Authors: Chang-Yeol Jung, Thien Binh Nguyen

Published in: Journal of Scientific Computing | Issue 2/2015

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Abstract

A semi-analytical method is developed based on conventional integrating factor (IF) and exponential time differencing (ETD) schemes for stiff problems. The latter means that there exists a thin layer with a large variation in their solutions. The occurrence of this stiff layer is due to the multiplication of a very small parameter \(\varepsilon \) with the transient term of the equation. Via singular perturbation analysis, an analytic approximation of the stiff layer, which is called a corrector, is sought for and embedded into the IF and ETD methods. These new schemes are then used to approximate the non-stiff part of the solution. Since the stiff part is resolved analytically by the corrector, the new method outperforms the conventional ones in terms of accuracy. In this paper, we apply our new method for both problems of ordinary differential equations and some partial differential equations.

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Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)CrossRefMATHMathSciNet

Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2–3), 151–167 (1997)CrossRefMATHMathSciNet

Ashi, H.: Numerical methods for stiff systems, Ph.D. thesis, the University of Nottingham (2008)

Aziz, et al. Z.A.: Fourth-order time stepping for stiff PDEs via integrating factor. Adv. Sci. Lett. 19(1), 170–173 (2013)

Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, Mineola, NY (2001)MATH

Cash, J.R.: Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. Proc. R. Soc. Lond. A. 459, 797–815 (2003)CrossRefMATHMathSciNet

Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, Berlin (1988)CrossRef

Cox, S.M., Matthews, P.C.: Exponential time differencing for stiff systems. J. Comput. Phys. 176, 430–455 (2002)CrossRefMATHMathSciNet

Weinan, E., Engquist, Bjorn, Li, Xiantao, Ren, Weiqing, Vanden-Eijnden, Eric: The heterogeneous multiscale method: a review. Commun. Comput. Phys. 2(3), 367–450 (2007)MATHMathSciNet

10.

Engquist, B., Tsai, Y.-H.: Heterogeneous multiscale methods for stiff ordinary differential equations. Math. Comput. 74(252), 1707–1742 (2005)CrossRefMATHMathSciNet

11.

Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, UK (1996)MATH

12.

Holmes, M.H.: Introduction to Perturbation Methods. Springer, New York (1995)CrossRefMATH

13.

Han, H., Kellogg, R. B.: A method of enriched subspaces for the numerical solution of a parabolic singular perturbation problem. In: Computational and Asymptotic Methods for Boundary and Interior Layers, Boole Press Conf. Ser. 4, Dublin, pp. 46–52 (1982)

14.

Hyman, J.M., Nicolaenko, B.: The Kuramoto-Sivashinsky equation: a bridge between PDE’s and dynamical systems. Physica D 18, 113–126 (1986). North-Holland, AmsterdamCrossRefMATHMathSciNet

15.

Hoz, F.D.L., Vadillo, F.: An exponential time differencing method for the nonlinear Schrödinger equation. Comput. Phys. Commun. 179, 449–456 (2008)CrossRefMATH

16.

Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems. Springer, Berlin (1996)CrossRefMATH

17.

Johnson, R.S.: Singular Perturbation Theory. Springer Science+Business Media Inc, New York (2005)MATH

18.

Jung, C.: Finite elements scheme in enriched subspaces for singularly perturbed reaction-diffusion problems on a square domain. Asymptot. Anal. 57, 41–69 (2008)MATHMathSciNet

19.

Jin, S., Levermore, C.D.: Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126, 449–467 (1996)CrossRefMATHMathSciNet

20.

Jung, C., Nguyen, T.B.: Semi-analytical numerical methods for convection-dominated problems with turning points. Int. J. Numer. Anal. Model. 10(2), 314–332 (2013)MATHMathSciNet

21.

Jung, C., Nguyen, T.B.: New time differencing methods for spectral methods (submitted)

22.

Jung, C., Temam, R.: Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point. J. Math. Phys. 48, 065301 (2007)CrossRefMathSciNet

23.

Jung, C., Temam, R.: Finite volume approximation of one-dimensional stiff convection-diffusion equations. J. Sci. Comput. 41(3), 384–410 (2009)CrossRefMATHMathSciNet

24.

Jin, S., Xin, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48(3), 235–276 (1995)CrossRefMATHMathSciNet

25.

Kevorkian, J., Cole, J.D.: Multiple Scale and Singular Perturbation Methods. Springer, Berlin (1996)CrossRefMATH

26.

Kellogg, R.B., Stynes, M.: Layers and corner singularities in singularly perturbed elliptic problems. BIT 48(2), 309–314 (2008)CrossRefMATHMathSciNet

27.

Kevrekidis, I.G., Samaey, G.: Equation-free multiscale computation: algorithms and applications. Annu. Rev. Phys. Chem. 60, 321–344 (2009)CrossRef

28.

Kassam, A.-K., Trefethen, L.N.: Fourth-order time-stepping for stiff PDEs. SIAM J. Sci. Comput. 26(4), 1214–1233.

29.

Krogstad, S.: Generalized integrating factors methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)CrossRefMATHMathSciNet

30.

Marion, M., Temam, R.: Nonlinear Galerkin methods. SIAM J. Numer. Anal. 26(5), 1139–1157 (1989)CrossRefMATHMathSciNet

31.

Mai-Duy, N., Pan, D., Phan-Thien, N., Khoo, B.C.: Dissipative particle dynamics modeling of low Reynolds number incompressible flows. J. Rheol. 57, 585 (2013)CrossRef

32.

Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of the Kuramoto-Sivashisky equations: nonlinear stability and attractors. Physica D 16, 155–183 (1985). North-Holland, AmsterdamCrossRefMATHMathSciNet

33.

O’Malley, R.E.: Singularly perturbed linear two-point boundary value problems. SIAM Rev. 50(3), 459–482 (2008)CrossRefMATHMathSciNet

34.

Shih, S., Kellogg, R.B.: Asymptotic analysis of a singular perturbation problem. SIAM J. Math. Anal. 18, 1467–1511 (1987)CrossRefMATHMathSciNet

35.

Stynes, M.: Steady-state convection-diffusion problems. Acta Numer. 14, 445–508 (2005)CrossRefMATHMathSciNet

36.

Trefethen, L.N.: Spectral methods in Matlab. Society for Industrial and Applied Mathematics, Philadelphia (2000)CrossRefMATH

37.

Tao, M., Owhadi, H., Marsden, J.E.: Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul. 8(4), 1269–1324 (2010)CrossRefMATHMathSciNet

38.

Wasow, W.: Linear Turning Point Theory. Spinger, New York (1985)CrossRefMATH

Title: Semi-analytical Time Differencing Methods for Stiff Problems
Authors: Chang-Yeol Jung
Thien Binh Nguyen
Publication date: 01-05-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 2/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-014-9897-y

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