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Journal of Scientific Computing

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08-09-2018

Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems

Authors: Qiang Du, Lili Ju, Jianfang Lu

Published in: Journal of Scientific Computing | Issue 3/2019

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Abstract

In this paper we first present stability and error analysis of the fully discrete numerical schemes for general dissipative systems, in which the implicit Runge–Kutta (IRK) method is adopted for time integration. Under suitable conditions on the IRK time stepping method that we refer as the total stability, a priori error estimates can be simultaneously obtained. Then we apply such time-marching techniques and analysis framework to one-dimensional time-dependent nonlocal diffusion problems, together with the discontinuous Galerkin method being used for spatial discretization. Unconditional stability of approximations of both primal and auxiliary variables and the priori error estimates for the corresponding fully discrete systems are proved, and the results indicate the schemes are asymptotically compatible. In addition, long time asymptotic behavior of the approximate solutions is also investigated. Various numerical experiments are finally performed to verify the theoretical results.

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Bates, P.W., Chmaj, A.: An integrodifferential model for phase transitions: stationary solutions in higher space dimensions. J. Stat. Phys. 95, 1119–1139 (1999)MathSciNetCrossRefMATH

Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, London (2008)CrossRefMATH

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)MATH

Cheng, Y., Zhang, Q.: Local analysis of local discontinuous Galerkin method with generalized alternating numerical flux for one-dimensional singularity perturbed problem. J. Sci. Comput. 72, 1–28 (2017)MathSciNetCrossRef

Cheng, Y., Zhang, Q.: Local analysis of the fully discrete local discontinuous Galerkin method for the time-dependent singularly perturbed problem. J. Comput. Math. 35, 265–288 (2017)MathSciNetCrossRefMATH

Crouzeix, M., Hundsdorfer, W.H., Spijker, M.N.: On the existence of solutions to the algebraic equations in implicit Runge–Kutta methods. BIT Numer. Math. 23, 84–91 (1983)MathSciNetCrossRefMATH

Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. III: One dimensional systems. J. Computat. Phys. 84, 90–113 (1989)MathSciNetCrossRefMATH

Cockburn, B., Shu, C.-W.: The Runge–Kutta local projection \(P^1\)-discontinuous-Galerkin finite element method for scalar conservation laws. Math. Model. Numer. Anal. 25, 337–361 (1991)MathSciNetCrossRefMATH

Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws. II: General framework. Math. Comput. 52, 411–435 (1989)MATH

10.

Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin finite element method for conservation laws. V: Multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)MathSciNetCrossRefMATH

11.

Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time-dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetCrossRefMATH

12.

Dekker, K.: Error bounds for the solution to the algebraic equations in Runge–Kutta methods. BIT Numer. Math. 24, 347–356 (1984)MathSciNetCrossRefMATH

13.

Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54, 667–696 (2012)MathSciNetCrossRefMATH

14.

Du, Q., Ju, L., Lu, J.: A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems. Math. Comput. (2017). https://doi.org/10.1090/mcom/3333 MATH

15.

Du, Q., Ju, L., Tian, L., Zhou, K.: A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models. Math. Comput. 82, 1889–1922 (2013)MathSciNetCrossRefMATH

16.

Dong, B., Shu, C.-W.: Analysis of a local discontinuous Galerkin method for linear time-dependent fourth-order problems. SIAM J. Numer. Anal. 47, 3240–3268 (2009)MathSciNetCrossRefMATH

17.

Du, Q., Yang, J.: Asymptotically compatible Fourier spectral approximations of nonlocal Allen–Cahn equations. SIAM J. Numer. Anal. 54, 1899–1919 (2016)MathSciNetCrossRefMATH

18.

Eftimie, R., De Vries, G., Lewis, M.A.: Complex spatial group patterns result from different animal communication mechanisms. Proc. Natl. Acad. Sci. 104, 6974–6979 (2007)MathSciNetCrossRefMATH

19.

Epshteyn, Y., Izmirlioglu, A.: Fully discrete analysis of a discontinuous finite element method for the Keller–Segel chemotaxis model. J. Sci. Comput. 40, 211–256 (2009)MathSciNetCrossRefMATH

20.

Fife, P.: Some nonclassical trends in parabolic and parabolic-like evolutions. In: Kirkilionis, M., Krömker, S., Rannacher, R., Tomi, F. (eds.) Trends in Nonlinear Analysis, pp. 153–191. Springer, Berlin (2003)CrossRef

21.

Ferracina, L., Spijker, M.N.: Strong stability of singly-diagonally-implicit Runge–Kutta methods. Appl. Numer. Math. 58, 1675–1686 (2008)MathSciNetCrossRefMATH

22.

Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)MathSciNetCrossRefMATH

23.

Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)MathSciNetCrossRefMATH

24.

Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, New York (1993)MATH

25.

Hochbruck, M., Pažur, T.: Implicit Runge–Kutta methods and discontinuous Galerkin discretizations for linear Maxwell’s equations. SIAM J. Numer. Anal. 53, 485–507 (2015)MathSciNetCrossRefMATH

26.

Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. Springer, New York (1991)CrossRefMATH

27.

Ketcheson, D.I., Macdonald, C.B., Gottlieb, S.: Optimal implicit strong stability preserving Runge–Kutta methods. Appl. Numer. Math. 59, 373–392 (2009)MathSciNetCrossRefMATH

28.

Luo, J., Shu, C.-W., Zhang, Q.: A priori error estimates to smooth solutions of the third order Runge–Kutta discontinuous Galerkin method for symmetrizable system of conservation laws. Math. Model. Numer. Anal. 49, 991–1018 (2015)MathSciNetCrossRefMATH

29.

Levy, D., Tadmor, E.: From semidiscrete to fully discrete: stability of Runge–Kutta schemes by the energy method. SIAM Rev. 40, 40–73 (1998)MathSciNetCrossRefMATH

30.

Rosasco, L., Belkin, M., Vito, E.D.: On learning with integral operators. J. Mach. Learn. Res. 11, 905–934 (2010)MathSciNetMATH

31.

Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)MathSciNetCrossRefMATH

32.

Silling, S.A., Lehoucq, R.B.: Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168 (2010)CrossRef

33.

Sun, Z., Shu, C.-W.: Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations. Ann. Math. Sci. Appl. 2, 255–284 (2017)MathSciNetMATH

34.

Tadmor, E.: From semidiscrete to fully discrete: stability of Runge–Kutta schemes by the energy method. II. In: Estep, D., Tavener, S. (eds.) Collected Lectures on the Preservation Of Stability Under Discretization, Proceedings of the Workshop on the Preservation of Stability Under Discretization, Colorado State University, Fort Collins, CO (2001); Proceedings in Applied Mathematics, vol. 109, pp. 25–49. SIAM (2002)

35.

Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)MathSciNetCrossRefMATH

36.

Tian, X., Du, Q.: Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52, 1641–1665 (2014)MathSciNetCrossRefMATH

37.

Tian, X., Du, Q.: Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal. 53, 762–781 (2015)MathSciNetCrossRefMATH

38.

Tian, H., Ju, L., Du, Q.: Nonlocal convection–diffusion problems and finite element approximations. Comput. Methods Appl. Mech. Eng. 289, 60–78 (2015)MathSciNetCrossRefMATH

39.

Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for advection–diffusion problems. SIAM J. Numer. Anal. 53, 206–227 (2015)MathSciNetCrossRefMATH

40.

Wang, H., Shu, C.-W., Zhang, Q.: Stability analysis and error estimates of local discontinuous Galerkin methods with implicit–explicit time-marching for nonlinear convection–diffusion problems. Appl. Math. Comput. 272, 237–258 (2016)MathSciNetMATH

41.

Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for nonlinear Schrödinger equations. J. Comput. Phys. 205, 72–97 (2005)MathSciNetCrossRefMATH

42.

Xia, Y., Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for the Cahn–Hilliard type equations. J. Comput. Phys. 227, 472–491 (2007)MathSciNetCrossRefMATH

43.

Yan, J., Shu, C.-W.: A local discontinuous Galerkin method for KdV type equations. SIAM J. Numer. Anal. 40, 769–791 (2002)MathSciNetCrossRefMATH

44.

Yan, J., Shu, C.-W.: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17, 27–47 (2002)MathSciNetCrossRefMATH

45.

Zhang, Q., Shu, C.-W.: Error estimates to smooth solution of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641–666 (2004)MathSciNetCrossRefMATH

46.

Zhang, Q., Shu, C.-W.: Stability analysis and a priori error estimate to the third order explicit Runge–Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal. 48, 1038–1063 (2010)MathSciNetCrossRefMATH

Title: Analysis of Fully Discrete Approximations for Dissipative Systems and Application to Time-Dependent Nonlocal Diffusion Problems
Authors: Qiang Du
Lili Ju
Jianfang Lu
Publication date: 08-09-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2019
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0815-6

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