Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
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IMPLICIT-EXPLICIT SECOND DERIVATIVE LMM FOR STIFF ORDINARY DIFFERENTIAL EQUATIONS

  • OGUNFEYITIMI, S.E. (ADVANCED RESEARCH LABORATORY, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF BENIN) ;
  • IKHILE, M.N.O. (ADVANCED RESEARCH LABORATORY, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF BENIN)
  • Received : 2021.09.27
  • Accepted : 2021.12.21
  • Published : 2021.12.25
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Abstract

The interest in implicit-explicit (IMEX) integration methods has emerged as an alternative for dealing in a computationally cost-effective way with stiff ordinary differential equations arising from practical modeling problems. In this paper, we introduce implicit-explicit second derivative linear multi-step methods (IMEX SDLMM) with error control. The proposed IMEX SDLMM is based on second derivative backward differentiation formulas (SDBDF) and recursive SDBDF. The IMEX second derivative schemes are constructed with order p ranging from p = 1 to 8. The methods are numerically validated on well-known stiff equations.

Keywords

References

  1. W. Hundsdorfer and J.G. Verwer: Numerical solution of time-dependent advection-diffusion-reaction equations-springer series in computational mathematics. Springer 33 (2003), 386-387.
  2. S.J. Ruuth: Implicit-explicit methods for reaction-diffusion problems in pattern formation. Journal of Mathematical Biology 34 (1995), 148-176. https://doi.org/10.1007/BF00178771
  3. J.G. Verwer, J.G. Blom and W. Hundsdorfer: An implicit-explicit approach for atmospheric transport-chemistry problems. Appl Numer Math 20 (1996), 191-209. https://doi.org/10.1016/0168-9274(95)00126-3
  4. J.R. Cash: Split linear multistep Methods for the numerical Integration of Stiff Differential System. Numer Math 42 (1983), 299-310. https://doi.org/10.1007/BF01389575
  5. M. Crouzeix: Une methode multipas implicite-explicite pour l'approximation des equations d'evolution paraboliques. Numer Math 35 (1980), 257-276. https://doi.org/10.1007/BF01396412
  6. J.M. Varah: Stability restriction on second order, three-level finite-difference schemes for parabolic equations. SIAM J Numer Anal 17 (1980), 300-309. https://doi.org/10.1137/0717025
  7. G. Akrivis and Y. Smyrlis: Implicit-explicit BDF methods for the Kuramoto-Sivashinsky equation. Appl Numer Math 51 (2004), 151-169. https://doi.org/10.1016/j.apnum.2004.03.002
  8. U.M. Ascher, S.J. Ruuth and B.T.R. Wetton: Implicit-explicit methods for time-dependent partial differential equations. SIAM J Numer Anal 32 (1995), 797-823. https://doi.org/10.1137/0732037
  9. J. Frank, W. Hundsdorfer, J. Verwer: On the stability of implicit-explicit linear multistep methods. Appl Numer. Math. 25 (1997), 193-205. https://doi.org/10.1016/S0168-9274(97)00059-7
  10. S.E. Ogunfeyitimi: Implicit-explicit second derivative linear multi-step methods for stiff ordinary differential equations. Master thesis, University of Benin, Benin, Nigeria (2016).
  11. S.E. Ogunfeyitimi and M.N.O. Ikhile: Implicit-explicit methods based on recursively derived second derivative LMM. J. Nigerian assoc of maths phys 38 (2016), 57-66.
  12. K.M. Owolabi and K.C. Patidar: Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology. J Appl Math and Comp 240 (2014), 30-50. https://doi.org/10.1016/j.amc.2014.04.055
  13. W. Hundsdorfer and J.G. Verwer: A note on splitting errors for advection-reaction equations. Appl Num Math 18 (1995), 191-199. https://doi.org/10.1016/0168-9274(95)00069-7
  14. W. Hundsdorfer and S.J. Ruuth: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J Comput Phys 225 (2007), 2016-2042. https://doi.org/10.1016/j.jcp.2007.03.003
  15. U.M. Ascher, S.J. Ruuth and R.J. Spiteri: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl Numer Math 25 (1997), 151-167. https://doi.org/10.1016/S0168-9274(97)00056-1
  16. M.P. Calvo, Jde.Frutos and J. Novo: Linearly implicit Runge-Kutta methods for advection-diffusion-reaction problems. Appl Numer Math 37 (2001), 535-549. https://doi.org/10.1016/S0168-9274(00)00061-1
  17. C.A. Kennedy and M.H. Carpenter: Additive Runge-Kutta schemes for convection-diffiusion-reaction equations. Appl Numer Math 44 (2003), 139-181. https://doi.org/10.1016/S0168-9274(02)00138-1
  18. L. Pareschi and G. Russo: Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations. In: Recent trends in numerical analysis. Adv Theory Comput Math 3 (2001), 269-288.
  19. L. Pareschi and G. Russo: Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J Sci Comput 25 (2005), 129-155. https://doi.org/10.1007/BF02728986
  20. X. Zhong: Additive semi-implicit Runge-Kutta methods for computing high-speed non equilibrium reactive flows. J Compo Phys 128 (1996), 19-31. https://doi.org/10.1006/jcph.1996.0193
  21. G. Izzo and Z. Jackiewicz: Highly stable implicit-explicit Runge-Kutta methods. Appl Numer Math 113 (2017), 71-92. https://doi.org/10.1016/j.apnum.2016.10.018
  22. O. Knoth and R. Wolke: Implicit-explicit Runge-Kutta methods for computing atmospheric reactive flows. Appl Numer Math 28 (1998), 327-341. https://doi.org/10.1016/S0168-9274(98)00051-8
  23. H. Zhang and A. Sandu: A second-order diagonally-implicit-explicit multistage integration method. Procedia CS 9 (2012), 1039--1046.
  24. E. Zharovski and A. Sandu: A class of implicit-explicit two-step Runge-Kutta methods. Tech. Rep. TR-12-08, Computer Science, Virginia.(2012)
  25. A. Cardone, Z. Jackiewicz, A. Sandu and H. Zhang: Extrapolation based implicit-explicit general linear methods. Numer Algorithms 65 (2014), 377-399. https://doi.org/10.1007/s11075-013-9759-y
  26. H. Zhang and A. Sandu: Partitioned and implicit-explicit general linear methods for ordinary differential equations. (2013) http://arxiv.org/abs/1302.2689.
  27. M. Bras, G. Izzo and Z. Jackiewicz: Accurate implicit-explicit general linear methods with inherent Runge-Kutta stability. J Sci Comput 70 (2017), 1105-1143. https://doi.org/10.1007/s10915-016-0273-y
  28. H. Zhang, A. Sandua and S. Blaise: High order implicit-explicit general linear methods with optimized stability regions. SIAM J Sci Comput 38 (2016), A1430-A1453. https://doi.org/10.1137/15M1018897
  29. M. Schneider, J. Lang and W. Hundsdorfer: Extrapolation-based super convergent implicit-explicit Peer methods with A-stable implicit part. J Comp Phys 367 (2018), 121-133. https://doi.org/10.1016/j.jcp.2018.04.006
  30. B. Soleimani and R. Weiner: Super-convergent IMEX Peer methods. Appl Numer Math 130 (2018), 70-85. https://doi.org/10.1016/j.apnum.2018.03.014
  31. M. Schneider, J. Lang, R. Weiner, Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes. J Comput Appl Math.(2019) doi:10.1016/j.cam.2019.112501
  32. S.E. Ogunfeyitimi and M.N.O. Ikhile: Second derivative generalized extended backward differentiation formulas for stiff problems. J Korean Soc Ind Appl Math 23 (2019), 179-202. Doi.org/10.12941/jksiam.2019.23.179
  33. S.E. Ogunfeyitimi and M.N.O. Ikhile: Multi-block boundary value methods for ordinary differential and differential algebraic equation. J Korean Soc Ind Appl Math 24 (2020), 243-291. https://doi.org/10.12941/jksiam.2020.24.243
  34. S.E. Ogunfeyitimi and M.N.O. Ikhile: Multi-block Generalized Adams-Type Integration Methods for Differential Algebraic Equations. Int. J. Appl. Comput. Math 7, 197 (2021). https://doi.org/10.1007/s40819-021-01135-x
  35. S.E Ogunfeyitimi and M.N.O Ikhile: Generalized second derivative linear multistep methods based on the methods of Enright. Int. J. Appl. and Comput. Math 6, 76 (2020). https://doi.org/10.1007/s40819-020-00827-0
  36. P.O. Olatunji and M.N.O Ikhile: Strongly regular general linear methods J. Sci. Comp. 82, 7 (2020), 1-30. Doi.org/10.1007/s10915-019-01107-w
  37. P.O. Olatunji, M.N.O Ikhile and R.I Okuonghae: Nested second derivative to-ste Runge-Kutta methods. Int. J. Appl. Comput. Math 7, 249 (2021). https://doi.org/10.1007/s40819-021-0119-1
  38. A. Prothero and Robinson A: On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations. Math Comp 28 (1974), 145-162. https://doi.org/10.1090/S0025-5718-1974-0331793-2
  39. C.F. Curtiss and J.O. Hirschfelder: Integration of stiff equations, Proceeding of the National Academy of Sciences. 38 (1952), 235-243. https://doi.org/10.1073/pnas.38.3.235
  40. S.O. Fatunla: Numerical methods for initial value problems in ordinary differential equations. Academic Press Inc New York. (1988)
  41. J.D. Lambert : Computational methods in ordinary differential equations Wiley, New York, 1973.
  42. E. Hairer and G. Wanner: Solving ordinary differential equation II: stiff and differential-algebraic problems. 2nd rev. ed., Springer-Verlag, New York.(1996)
  43. Z. Jackiewicz: General linear method for ordinary differential equations. John Wiley and Sons, Inc, Hoboken, New Jersey.(2009)
  44. D.J. Higham and L.N Trefethen: Stiffness of ODEs. BIT Numer Math 33 (1993), 285-303. https://doi.org/10.1007/BF01989751
  45. G. Dahlquist: Convergence and stability in the numerical integration of ordinary differential equations. Math Scand 4 (1956), 33-53. https://doi.org/10.7146/math.scand.a-10454
  46. W.H. Enright: Second derivative multistep methods for stiff ordinary differential equations. SIAM J Numer Anal 11 (1974), 321-331. https://doi.org/10.1137/0711029
  47. J.R. Cash: Second derivative extended backward differentiation formulas for the numerical integration of stiff systems. SIAM J Numer Anal 18 (1981), 21-36. https://doi.org/10.1137/0718003
  48. A.D. Jorgenson, Unconditional stability of a Crank-Nicolson Adam-Bashforth2 implicit-explicit numerical method. Inter. J Numer Analysis and Modeling Series B 5 (2014), 171-187.
  49. K. Gustasfson, M. Lundh and G.Soderlind: A PI step size control for the numerical solution of ordinary differential equations. BIT 28 (1988), 270-287. https://doi.org/10.1007/BF01934091
  50. O. Nevanlinna, R. Jeltsch, Stability and accuracy of time discretizations for initial value problems. Numerische Mathematik 40 (1982), 245-296. https://doi.org/10.1007/BF01400542