Top

Journal of Scientific Computing

Published in:

28-01-2016

High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations

Authors: Sebastiano Boscarino, Francis Filbet, Giovanni Russo

Published in: Journal of Scientific Computing | Issue 3/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The main purpose of the paper is to show how to use implicit–explicit Runge–Kutta methods in a much more general context than usually found in the literature, obtaining very effective schemes for a large class of problems. This approach gives a great flexibility, and allows, in many cases the construction of simple linearly implicit schemes without any Newton’s iteration. This is obtained by identifying the (possibly linear) dependence on the unknown of the system which generates the stiffness. Only the stiff dependence is treated implicitly, then making the whole method much simpler than fully implicit ones. The resulting schemes are denoted as semi-implicit R–K. We adopt several semi-implicit R–K methods up to order three. We illustrate the effectiveness of the new approach with many applications to reaction–diffusion, convection diffusion and nonlinear diffusion system of equations.

previous article A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

next article High Order Finite Difference Methods for the Wave Equation with Non-conforming Grid Interfaces

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Caflisch, R.E., Jin, S., Russo, G.: Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34(1), 246–281 (2001)MathSciNetCrossRefMATH

Carpenter, M.H., Kennedy, C.A.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44, 139–181 (2003)MathSciNetCrossRefMATH

Chen, C.Q., Levermore, C.D., Liu, T.P.: Hyperbolic conservation laws with relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994)MathSciNetCrossRefMATH

Durran, D.R., Blossey, P.N.: Implicit–explicit multistep methods for fast-wave–slow-wave problems. Mon. Wea. Rev. 140, 1307–1325 (2012)CrossRef

Jin, S.: Runge–Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 122, 51–67 (1995)MathSciNetCrossRefMATH

Pareschi, L., Russo, G.: Implicit–explicit Runge–Kutta schemes for stiff systems of differential equations. In: Trigiante, D. (ed.) Recent Trends in Numerical Analysis, Advances Theory Computational Mathematics, vol. 3, pp. 269–288. Nova Sci. Publ., Huntington, NY (2001)

Pareschi, L., Russo, G.: Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxations. J. Sci. Comput. 25, 129–155 (2005)MathSciNetMATH

Weller, H., Lock, S.J., Wood, N.: Runge–Kutta IMEX schemes for the horizontally explicit/vertically implicit (HEVI) solution of wave equations. J. Comput. Phys. 252, 365–381 (2013)MathSciNetCrossRef

Hairer, E., Wanner, G.: Solving ordinary differential equation. II. Stiff and differential algebraic problems, Springer Series in Computational Mathematics, 14, Springer (second revised edition 1996), corrected second printing 2002

10.

Higueras, I., Mantas, J.M., Roldán, T.: Design and implementation of predictors for additive semi-implicit Runge–Kutta methods. SIAM J. Sci. Comput. 31(3), 2131–2150 (2009)MathSciNetCrossRefMATH

11.

Zhong, X.: Additive semi-implicit Runge–Kutta methods for computing high-speed non equilibrium reactive flows. J. Comput. Phys. 128, 19–31 (1996)MathSciNetCrossRefMATH

12.

Boscarino, S., LeFloch, P.-G., Russo, G.: High-order asymptotic-preserving methods for fully non linear relaxation problems. SIAM J. Sci. Comput. 36, A377–A395 (2014)MathSciNetCrossRefMATH

13.

Berthon, C., LeFloch, P.-G., Turpault, R.: Late-time relaxation limits of nonlinear hyperbolic systems. A general framework. Math. Comput. 82, 831–860 (2012)MathSciNetCrossRefMATH

14.

Smereka, P.: Semi-implicit level set methods for curvature and surface diffusion motion. J. Sci. Comput. 19(1–3), 439–456 (2003)MathSciNetCrossRefMATH

15.

Filbet, F., Jin, S.: A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229, 7625–7648 (2010)MathSciNetCrossRefMATH

16.

Giraldo, F.X., Restelli, M., Läuter, M.: Semi-implicit formulations of the Navier–Stokes equations: application to nonhydrostatic atmosphereric modeling. SIAM J. Sci. Comput. 32(6), 3394–3425 (2010)MathSciNetCrossRefMATH

17.

Giraldo, F.X., Kelly, J.F., Costantinescu, E.M.: Implicit–explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA). SIAM J. Sci. Comput. 35(5), B1162–B1194 (2013)MathSciNetCrossRefMATH

18.

Araújo, A., Barbeiro, S., Serranho, P.: Stability of finite difference schemes for complex diffusion processes. SIAM J. Numer. Anal. 50, 1284–1296 (2012)MathSciNetCrossRefMATH

19.

Boscarino, S., Russo, G.: High-order asymptotic-preserving methods for nonlinear relaxation from hyperbolic systems to convection–diffusion equations. In: Submitted to Proceedings, High Order Nonlinear Numerical Methods for Evolutionary PDEs (HONOM 2013), March 18–22, Bordeaux (2013)

20.

Boscarino, S., Bürger, R., Mulet, Pep, Russo, G., Villada, L.M.: Linearly implicit IMEX Runge–Kutta methods for a class of degenerate convection–diffusion problems. SIAM J. Sci. Comput. 37, B305–B331 (2015)MathSciNetCrossRefMATH

21.

Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equation. I. Non-stiff Problems, Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993)MATH

22.

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

23.

Boscarino, S.: Error analysis of IMEX Runge–Kutta methods derived from differential algebraic systems. SIAM J. Numer. Anal. 45, 1600–1621 (2007)MathSciNetCrossRefMATH

24.

Boscarino, S.: On an accurate third order implicit–explicit Runge–Kutta method for stiff problems. Appl. Numer. Math. 59, 1515–1528 (2009)MathSciNetCrossRefMATH

25.

Boscarino, S., Russo, G.: Flux-explicit IMEX Runge–Kutta schemes for hyperbolic to parabolic relaxation problems. SIAM J. Numer. Anal. 51, 163–190 (2013)MathSciNetCrossRefMATH

26.

Hairer, E., Lubich, C., Roche, M.: Error of Runge–Kutta methods for stiff problems studied via differential algebraic equations. BIT Numer. Math. 28(3), 678–700 (1988)MathSciNetCrossRefMATH

27.

Filbet, F., Guo, R., Xu, Y.: Efficient high order semi-implicit time discretization and local discontinuous Galerkin methods for highly nonlinear PDEs. Preprint (2015)

28.

Filbet, F., Rodrigues, L.M.: Asymptotically stable particle-in-cell methods for the Vlasov–Poisson system with a strong external magnetic field. Preprint (2015)

29.

Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, vol. 31, 2nd edn. Springer, Berlin (2006)MATH

30.

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

31.

Boscarino, S., Russo, G.: On a class of uniformly accurate IMEX Runge–Kutta schemes and application to hyperbolic systems with relaxation. SIAM J. Sci. Comput. 31, 1926–1945 (2009)MathSciNetCrossRefMATH

32.

Higueras, I., Roldán, T.: Construction of additive semi-implicit Runge–Kutta methods with low-storage requirements. J. Sci. Comput. (2015). doi:10.1007/s10915-015-0116-2

33.

Filbet, F., Rambaud, A.: Analysis of an asymptotic preserving scheme for relaxation systems. ESAIM Math. Model. Numer. Anal. 47, 609–633 (2013)MathSciNetCrossRefMATH

34.

Zhang, K., Wong, J.C.F., Zhang, R.: Second-order implicit–explicit scheme for the Gray–Scott model. J. Comput. Appl. Math. 213, 559–581 (2008)MathSciNetCrossRefMATH

35.

Carrillo, J.A., Laurençot, Ph, Rosado, J.: Fermi–Dirac–Fokker–Planck equation: well-posedness and long-time asymptotics. J. Differ. Equ. 247, 2209–2234 (2009)MathSciNetCrossRefMATH

36.

Carrillo, J.A., Rosado, J., Salvarani, F.: 1D nonlinear Fokker–Planck equations for fermions and bosons. Appl. Math. Lett. 21, 148–154 (2008)MathSciNetCrossRefMATH

37.

Bessemoulin-Chatard, M., Filbet, F.: A finite volume scheme for nonlinear degenerate parabolic equations. SIAM J. Sci. Comput. 34, 559–583 (2012)MathSciNetCrossRefMATH

38.

Bertozzi, A.L.: The mathematics of moving contact lines in thin liquid films. Not. AMS 45, 689–697 (1998)MathSciNetMATH

39.

Pareschi, L., Russo, G., Toscani, G.: A kinetic approximation to Hele–Shaw flow. C. R. Math., Acad. Sci. Paris, Ser. I 338(2), 178–182 (2004)MathSciNetCrossRefMATH

40.

Bertozzi, A.L., Pugh, M.: The lubrication approximation for thin viscous films: regularity and long-time behavior of weak solutions. Commun. Pure Appl. Math. XLIX, 85–123 (1996)MathSciNetCrossRefMATH

41.

Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232 (2005)MathSciNetCrossRefMATH

42.

Xu, Y., Shu, C.W.: Local discontinuous Galerkin method for surface diffusion and Willmore flow of graphs. J. Sci. Comput. 40, 375–390 (2009)MathSciNetCrossRefMATH

Title: High Order Semi-implicit Schemes for Time Dependent Partial Differential Equations
Authors: Sebastiano Boscarino
Francis Filbet
Giovanni Russo
Publication date: 28-01-2016
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 3/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-016-0168-y

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 3/2016

Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows

Convergence Rate for the Ordered Upwind Method

Notes on RKDG Methods for Shallow-Water Equations in Canal Networks

A Posteriori Error Analysis of the Discontinuous Galerkin Method for Two-Dimensional Linear Hyperbolic Conservation Laws on Cartesian Grids

Point Source Super-resolution Via Non-convex $$L_1$$ L 1 Based Methods

Erratum to: Fast Computing of Conformal Mapping and Its Inverse of Bounded Multiply Connected Regions onto Second, Third and Fourth Categories of Koebe’s Canonical Slit Regions

Premium Partner