Accurate evaluation of divided differences for polynomial interpolation of exponential propagators

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator phi(hA) v, phi(z) = (e(z)-1)/z. In this way, it is possible to approximate.( hA) v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to "higher" order phi(k) functions, k >= 0

Comparison of software for computing the action of the matrix exponential

The implementation of exponential integrators requires the action of the matrix exponential and related functions of a large matrix. There are various methods in the literature for carrying out this task. In this paper we describe a new implementation of a method based on interpolation at Leja points. We numerically compare this method with others from the literature. As we are interested in exponential intergrators we choose the test examples from spatial discretization of time dependent partial differential equations in two and three space dimensions. The test matrices thus have large eigenvalues and can be nonnormal

A massively parallel exponential integrator for advection-diffusion models

This work considers the Real Leja Points Method (ReLPM) for the exponential integration of large-scale sparse systems of ODEs, generated by Finite Element or Finite Difference discretizations of 3-D advection-diffusion models. We present an efficient parallel implementation of ReLPM for polynomial interpolation of the matrix exponential propagators. A scalability analysis of the most important computational kernel inside the code, the parallel sparse matrix\u2013vector product, has been performed, as well as an experimental study of the communication overhead. As a result of this study an optimized parallel sparse matrix\u2013vector product routine has been implemented. The resulting code shows good scaling behavior even when using more than one thousand processors. The numerical results presented on a number of very large test cases gives experimental evidence that ReLPM is a reliable and efficient tool for the simulation of complex hydrodynamic processes on parallel architectures

Numerically stable formulas for a material point based explicit exponential integrator

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. &nbsp

Backward error analysis of polynomial approximations for computing the action of the matrix exponential

We describe how to perform the backward error analysis for the approximation of exp(A)v by p(s 121 A) s v, for any given polynomial p(x). The result of this analysis is an optimal choice of the scaling parameter s which assures a bound on the backward error, i.e. the equivalence of the approximation with the exponential of a slightly perturbed matrix. Thanks to the SageMath package expbea we have developed, one can optimize the performance of the given polynomial approximation. On the other hand, we employ the package for the analysis of polynomials interpolating the exponential function at so called Leja\u2013Hermite points. The resulting method for the action of the matrix exponential can be considered an extension of both Taylor series approximation and Leja point interpolation. We illustrate the behavior of the new approximation with several numerical examples

Eine flexible Klasse von local time stepping Verfahren

In dieser Dissertation werden numerische Zeitintegratoren für partielle Differentialgleichungen behandelt. Primär geht es um Integratoren für partielle Differentialgleichungen auf Gebieten, deren räumliche Diskretisierung eine lokale Verfeinerung erfordert. Untersucht werden die hiermit verbundenen Schwierigkeiten. Die Konstruktion, die Analyse und die Implementierung geeigneter Integratoren werden ebenfalls vorgestellt