## Abstract

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.

### Similar content being viewed by others

## Notes

As the algebra involved is overwhelming and error-prone, we have used the computer algebra system Maple to perform the simplifications and verifications. Thus, human intervention is dedicated to identify patterns and to discover abstract expressions such as \(x_\mathrm {p}\), \(x_\mathrm {t}\), \(x_\mathrm {q}\), etc.

In this form the difference of the independent variables appear symbolically as input to a function that could be evaluated to machine precision

## References

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

Caliari M, Ostermann A, Rainer S (2013) Meshfree exponential integrators. SIAM J Sci Comput 35(1):A431–A452. doi:10.1137/100818236

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

Gilbert F, Backus GE (1966) Propagator matrices in elastic wave and vibration problems. Geophysics 31(2):326–332. doi:10.1190/1.1439771

Higham NJ (2002) Accuracy and stability of numerical algorithms. Soc for Ind Appl Math doi:10.1137/1.9780898718027

Higham NJ (2008) Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics, Philadelphia

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

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

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

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

Kahan W, Darcy JD (1998) How java’s floating-point hurts everyone everywhere. ACM 1998 workshop on java for high-performance network computing. http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf. Online; Accessed Feb 5 2014

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

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

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

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

OpenCFD Ltd (ESI group): OpenFOAM–The open source CFD toolbox. http://www.openfoam.org/

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

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

Price JF (2006) Lagrangian and Eulerian representations of fluid flow: kinematics and the equations of motion. Woods Hole Oceanographic Institution. http://www.whoi.edu/science/PO/people/jprice/class/ELreps.pdf. Online; Accessed Dec 10 2012

Weisstein EW (2013) Cubic formula. From MathWorld–a Wolfram web resource. http://mathworld.wolfram.com/CubicFormula.html. Online; Accessed Oct 23 2013

## Acknowledgments

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.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

*Dedicated to Prof. Eugenio Oñate on the occasion of his 62nd birthday.*

## Rights and permissions

## About this article

### Cite this article

Nadukandi, P. Numerically stable formulas for a particle-based explicit exponential integrator.
*Comput Mech* **55**, 903–920 (2015). https://doi.org/10.1007/s00466-015-1142-5

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00466-015-1142-5

### Keywords

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