Numerically stable formulas for a material point based explicit exponential integrator

Abstract

We present numerically stable formulas for the analytical solution in the closed form of the so-called X-IVAS scheme in 3D. The X-IVAS scheme is a material point based explicit exponential integrator. An intermediate step in the X-IVAS scheme is the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes. This is what we refer to as particle tracing of streamlines and independent formulas for the same can be easily distilled from the ones presented for the X-IVAS scheme. The formulas involve functions of matrices which are defined using the corresponding Newton interpolating polynomial. The evaluation of these formulas is stable, i.e. a certain number of significant digits in the computed values are guaranteed to be exact. Using the double-precision floating-point arithmetic specified by the IEEE 754 standard, we obtain at least 10 significant decimal digits in the worst case scenarios. These scenarios involve fourth-order divided differences of the exponential function. Additionally, an optimal series approximation of divided differences is presented which is an essential part of the exposition.