Research output: Contribution to Journal/Magazine › Journal article › peer-review

Published

**A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.** / Mosbach, Sebastian; Turner, Amanda.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Mosbach, S & Turner, A 2009, 'A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.', *Computers and Mathematics with Applications*, vol. 57, no. 7, pp. 1157-1167. https://doi.org/10.1016/j.camwa.2009.01.020

Mosbach, S., & Turner, A. (2009). A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution. *Computers and Mathematics with Applications*, *57*(7), 1157-1167. https://doi.org/10.1016/j.camwa.2009.01.020

Mosbach S, Turner A. A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution. Computers and Mathematics with Applications. 2009 Apr;57(7):1157-1167. doi: 10.1016/j.camwa.2009.01.020

@article{b056f7b562c747c6bf73846f5cf587ec,

title = "A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.",

abstract = "We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order Runge–Kutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method.",

keywords = "Rounding errors, Markov jump processes, Numerical ODE solution, Limit theorem, Saddle fixed point",

author = "Sebastian Mosbach and Amanda Turner",

note = "The final, definitive version of this article has been published in the Journal, Computers and Mathematics with Applications 57 (7), 2009, {\textcopyright} ELSEVIER.",

year = "2009",

month = apr,

doi = "10.1016/j.camwa.2009.01.020",

language = "English",

volume = "57",

pages = "1157--1167",

journal = "Computers and Mathematics with Applications",

issn = "0898-1221",

publisher = "Elsevier Limited",

number = "7",

}

TY - JOUR

T1 - A quantitative probabilistic investigation into the accumulation of rounding errors in numerical ODE solution.

AU - Mosbach, Sebastian

AU - Turner, Amanda

N1 - The final, definitive version of this article has been published in the Journal, Computers and Mathematics with Applications 57 (7), 2009, © ELSEVIER.

PY - 2009/4

Y1 - 2009/4

N2 - We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order Runge–Kutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method.

AB - We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs) from a probabilistic viewpoint. We show that the accumulation of rounding errors results in a solution which is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and fourth order Runge–Kutta methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5 in order to demonstrate that the observed effects are not specific to a particular method.

KW - Rounding errors

KW - Markov jump processes

KW - Numerical ODE solution

KW - Limit theorem

KW - Saddle fixed point

U2 - 10.1016/j.camwa.2009.01.020

DO - 10.1016/j.camwa.2009.01.020

M3 - Journal article

VL - 57

SP - 1157

EP - 1167

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 7

ER -