Long-time behavior of numerical solutions to nonlinear fractional ODEs

Wang, Dongling; Xiao, Aiguo; Zou, Jun

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 1, pp. 335-358.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

In this work, we study the long time behavior, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By means of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grünwald–Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including convolution quadrature schemes based on classical second order BDF and product integration schemes based on quadratic interpolation approximation, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

[1] A.A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations. Differ. Equ. 46 (2010) 660–666. | DOI | MR | Zbl

[2] J.A.D. Applelby, I. Györi and D.W. Reynolds, On exact convergence rates for solutions of linear systems of Volterra difference equations. J. Diff. Equa. Appl. 12 (2006) 1257–1275. | DOI | MR | Zbl

[3] J.C. Butcher, A stability property of implicit Runge-Kutta methods. BIT Numer. Math. 15 (1975) 358–361. | DOI | Zbl

[4] J.C. Butcher, Thirty years of G-stability. BIT Numer. Math. 46 (2006) 479–489. | DOI | MR | Zbl

[5] J. Cao, C. Li and Y.Q. Chen, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II). Fract. Calc. Appl. Anal. 18 (2015) 735–761. | DOI | MR

[6] W. Cao, Z. Zhang and G.E. Karniadakis, Time-splitting schemes for fractional differential equations I: smooth solutions. SIAM J. Sci. Comput. 37 (2015) A1752–A1776. | DOI | MR

[7] W. Cao, F. Zeng, Z. Zhang and G.E. Karniadakis, Implicit-explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38 (2016) A3070–A3093. | DOI | MR

[8] J. Čermák, I. Györi and L. Nechvátal, On explicit stability conditions for a linear fractional difference system. Fract. Calc. Appl. Anal. 18 (2015) 651–672. | DOI | MR

[9] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Discrete Contin. Dyn. Syst. 277–285 (2007). | MR | Zbl

[10] E. Cuesta, C. Lubich and C. Palencia, Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75 (2006) 673–696. | DOI | MR | Zbl

[11] G. Dahlquist, Error analysis for a class of methods for stiff nonlinear initial value problems. In: Vol. 506 of Numerical Analysis, Lecture Notes in Mathematics. Springer Berlin Heidelberg (1975) 60–74. | MR | Zbl

[12] G. Dahlquist, G-stability is equivalent to A-stability. BIT Numer. Math. 18 (1978) 384–401. | DOI | Zbl

[13] K. Diethelm and N.J. Ford, Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2002) 229–248. | DOI | MR | Zbl

[14] K. Diethelm, N.J. Ford and A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29 (2002) 3–22. | DOI | MR | Zbl

[15] P.P.B. Eggermont, Uniform error estimates of Galerkin methods for monotone Abel-Volterra integral equations on the half-line. Math. Comput. 53 (1989) 157–189. | DOI | MR | Zbl

[16] L. Galeone and R. Garrappa, On multistep methods for differential equations of fractional order. Mediterr. J. Math. 3 (2006) 565–580. | DOI | MR | Zbl

[17] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations. Int. J. Comput. Math. 87 (2010) 2281–2290. | DOI | MR | Zbl

[18] R. Garrappa, Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math. Comput. Simul. 110 (2015) 96–112. | DOI | MR

[19] G.H. Gao, Z.Z. Sun and H.W. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259 (2014) 33–50. | DOI | MR

[20] E. Hairer and G. Wanner, Solving ordinary differential equations II, 2nd edition. In: Vol. 14 of Stiff and Differential-Algebraic Equations. Springer Series in Computational Mathematics. Springer, Berlin (1996). | MR | Zbl

[21] J.K. Hale, Asymptotic Behavior of Dissipative Systems. American Mathematical Society, New York (2010). | DOI | MR | Zbl

[22] A.T. Hill, Global dissipativity for A-stable methods. SIAM J. Numer. Anal. 34 (1997) 119–142. | DOI | MR | Zbl

[23] A.R. Humphries and A.M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems. SIAM J. Numer. Anal. 31 (1994) 1452–1485. | DOI | MR | Zbl

[24] B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA J. Numer. Anal. 36 (2016) 197–221. | MR

[25] B. Jin, R. Lazarov, V. Thomée and Z. Zhou, On nonnegativity preservation in finite element methods for subdiffusion equations. Math. Comput. 86 (2017) 2239–2260. | DOI | MR

[26] B. Jin, B. Li and Z. Zhou, Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39 (2017) A3129–A3152. | DOI | MR

[27] B. Jin, B. Li and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56 (2018) 1–23. | DOI | MR

[28] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Amsterdam (2006). | MR | Zbl

[29] N. Kopteva, Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88 (2019) 2135–2155. | DOI | MR

[30] H. Li, J. Cao and C. Li, High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (III). J. Comput. Appl. Math. 299 (2016) 159–175. | DOI | MR

[31] C.P. Li and F.R. Zhang, A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top. 193 (2011) 27–47. | DOI

[32] Y. Li, Y.Q. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45 (2009) 1965–1969. | DOI | MR | Zbl

[33] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007) 1533–1552. | DOI | MR | Zbl

[34] C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations. SIAM J. Sci. Comput. 38 (2016) A2699–A2724. | DOI | MR

[35] C. Lubich, On the stability of linear multistep methods for Volterra convolution equations. IMA J. Numer. Anal. 3 (1983) 439–465. | DOI | MR | Zbl

[36] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind. Math. Comput. 45 (1985) 463–469. | DOI | MR | Zbl

[37] D. Matignon. Stability results for fractional differential equations with applications to control processing. In: Vol. 2 of Computational Engineering in Systems Applications. IMACS, IEEE-SMC, Lille, France (1996) 963–968.

[38] O. Nevanlinna, On the numerical solutions of some Volterra equations on infinite intervals. Math.-Rev. Anal. Numr. Thor. Approx. 5 (1976) 31–57. | MR | Zbl

[39] I. Petras, Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation. Higher Education Press Beijing and Springer-Verlag, Berlin (2011). | DOI | Zbl

[40] I. Podlubny, Fractional Differential Equations, Academic Press, London (1998). | MR | Zbl

[41] Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56 (2006) 193–209. | DOI | MR | Zbl

[42] R. Temam, Infinite dimensional dynamical systems in mechanics and physics. In: Vol. 68 of Applied Mathematical Sciences. Springer-Verlag, Berlin (1998). | MR | Zbl

[43] D. Wang and A. Xiao, Dissipativity and contractivity for fractional-order systems. Nonlinear Dyn. 80 (2015) 287–294. | DOI | MR

[44] D. Wang and J. Zou, Dissipativity and contractivity analysis for fractional functional differential equations and their numerical approximations. SIAM J. Numer. Anal. 57 (2019) 1445–1470. | DOI | MR

[45] Y. Xing and Y. Yan, A higher order numerical method for time fractional partial differential equations with nonsmooth data. J. Comput. Phys. 357 (2018) 305–323. | DOI | MR

[46] D. Xu, Uniform $l^1$ behavior for time discretization of a Volterra equation with completely monotonic kernel II: Convergence. SIAM J. Numer. Anal. 46 (2008) 231–259. | DOI | MR | Zbl

[47] D. Xu, Decay properties for the numerical solutions of a partial differential equation with memory. J. Sci. Comput. 62 (2015) 146–178. | DOI | MR

[48] Y. Yan, M. Khan and N.J. Ford, An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data. SIAM J. Numer. Anal. 56 (2018) 210–227. | DOI | MR