Computational Mechanics

, Volume 55, Issue 5, pp 903–920 | Cite as

Numerically stable formulas for a particle-based explicit exponential integrator

  • Prashanth Nadukandi
Original Paper


Numerically stable formulas are presented for the closed-form analytical solution of the X-IVAS scheme in 3D. This scheme is a state-of-the-art particle-based explicit exponential integrator developed for the particle finite element method. Algebraically, this scheme involves two steps: (1) the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes and (2) the solution of line integrals of piecewise linear vector-valued functions along these tangent curves. Hence, the stable formulas presented here have general applicability, e.g. exact integration of trajectories in particle-based (Lagrangian-type) methods, flow visualization and computer graphics. The Newton form of the polynomial interpolation definition is used to express exponential functions of matrices which appear in the analytical solution of the X-IVAS scheme. The divided difference coefficients in these expressions are defined in a piecewise manner, i.e. in a prescribed neighbourhood of removable singularities their series approximations are computed. An optimal series approximation of divided differences is presented which plays a critical role in this methodology. At least ten significant decimal digits in the formula computations are guaranteed to be exact using double-precision floating-point arithmetic. The worst case scenarios occur in the neighbourhood of removable singularities found in fourth-order divided differences of the exponential function.


X-IVAS scheme Particle finite element method Explicit exponential integrators Tangent curves Closed-form analytical solutions Finite arithmetic Loss of significance Numerically stable formulas 

Mathematics Subject Classification

Primary: 65-04 Secondary: 34A05 39-04 65G30 70B05 76M10 76M28 



I thank Mr. Guillermo Casas-González (ORCiD: 0000-0002-1859-720X) for reading the manuscript in draft form and suggesting improvements. This study was partially supported by the SAFECON project of the European Research Council (European Commission) and the WAM-V project funded under the Navy Grant N62909-12-1-7101 issued by Office of Naval Research Global. The United States Government has a royalty-free license throughout the world in all copyrightable material contained herein.


  1. 1.
    Caliari M (2007) Accurate evaluation of divided differences for polynomial interpolation of exponential propagators. Computing 80(2):189–201. doi: 10.1007/s00607-007-0227-1 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Caliari M, Ostermann A, Rainer S (2013) Meshfree exponential integrators. SIAM J Sci Comput 35(1):A431–A452. doi: 10.1137/100818236 MathSciNetCrossRefGoogle Scholar
  3. 3.
    Diachin DP, Herzog JA (1997) Analytic streamline calculations on linear tetrahedra. In: 13th computational fluid dynamics conference. American Institute of Aeronautics and Astronautics, Reston, pp. 733–742 doi: 10.2514/6.1997-1975
  4. 4.
    Gilbert F, Backus GE (1966) Propagator matrices in elastic wave and vibration problems. Geophysics 31(2):326–332. doi: 10.1190/1.1439771 CrossRefGoogle Scholar
  5. 5.
    Higham NJ (2002) Accuracy and stability of numerical algorithms. Soc for Ind Appl Math doi: 10.1137/1.9780898718027
  6. 6.
    Higham NJ (2008) Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
  7. 7.
    Hochbruck M, Lubich C, Selhofer H (1998) Exponential integrators for large systems of differential equations. SIAM J Sci Comput 19(5):1552–1574. doi: 10.1137/S1064827595295337 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Idelsohn SR, Marti J, Becker P, Oñate E (2014) Analysis of multifluid flows with large time steps using the particle finite element method. Int J Numer Methods Fluids 75(9):621–644. doi: 10.1002/fld.3908
  9. 9.
    Idelsohn SR, Nigro N, Limache A, Oñate E (2012) Large time-step explicit integration method for solving problems with dominant convection. Comput Methods Appl Mech Eng 217–220, 168–185. doi: 10.1016/j.cma.2011.12.008
  10. 10.
    Idelsohn SR, Oñate E, Del Pin F (2004) The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int J Numer Methods Eng 61(7):964–989. doi: 10.1002/nme.1096 CrossRefGoogle Scholar
  11. 11.
    Kahan W, Darcy JD (1998) How java’s floating-point hurts everyone everywhere. ACM 1998 workshop on java for high-performance network computing. Online; Accessed Feb 5 2014
  12. 12.
    Kipfer P, Reck F, Greiner G (2003) Local exact particle tracing on unstructured grids. Comput Graphics Forum 22(2):133–142. doi: 10.1111/1467-8659.00655 CrossRefGoogle Scholar
  13. 13.
    McCurdy AC, Ng KC, Parlett BN (1984) Accurate computation of divided differences of the exponential function. Math Comput 43(168), 501–501. doi: 10.1090/S0025-5718-1984-0758198-0
  14. 14.
    Nielson GM, Jung IH (1999) Tools for computing tangent curves for linearly varying vector fields over tetrahedral domains. IEEE Trans Vis Comput Graphics 5(4), 360–372. doi: 10.1109/2945.817352
  15. 15.
    Oñate E, Idelsohn SR, Del Pin F, Aubry R (2004) The particle finite element method. An overview. Int J Comput Methods 1(2), 267–307. doi: 10.1142/S0219876204000204
  16. 16.
    OpenCFD Ltd (ESI group): OpenFOAM–The open source CFD toolbox.
  17. 17.
    Ostermann A, Thalhammer M, Wright WM (2006) A class of explicit exponential general linear methods. BIT Numer Math 46(2):409–431. doi: 10.1007/s10543-006-0054-3 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Parlett BN (1976) A recurrence among the elements of functions of triangular matrices. Linear Algebr Appl 14(2):117–121. doi: 10.1016/0024-3795(76)90018-5
  19. 19.
    Price JF (2006) Lagrangian and Eulerian representations of fluid flow: kinematics and the equations of motion. Woods Hole Oceanographic Institution. Online; Accessed Dec 10 2012
  20. 20.
    Weisstein EW (2013) Cubic formula. From MathWorld–a Wolfram web resource. Online; Accessed Oct 23 2013

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Centre Internacional de Mètodos Numèrics en Enginyeria (CIMNE)BarcelonaSpain

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