# Numerically stable formulas for a particle-based explicit exponential integrator

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## 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.

## 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## Mathematics Subject Classification

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

### 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.

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