Mitigating the integration error in numerical simulations of Newtonian systems.

Details Mitigating the integration error in numerical simulations of Newtonian systems. Mitigating the integration error in numerical simulations of Newtonian systems.

Oct 2006

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2006ijnme..68..267g&link_type=abstract

International Journal for Numerical Methods in Engineering, Volume 68, Issue 2, Pages: 267 - 297.

Statistics

Methodology

Oridinary Differential Equations, Initial Value Problems, Variation Of Parameters, Gauge Theory

Scientific paper

We introduce a method for mitigating the numerical integration errors of initial value problems. We propose a methodology for constructing an optimal state-space representation that gives minimum numerical truncation error, and in this sense, is the optimal state-space representation for modeling given phase-space dynamics. To that end, we utilize a simple transformation of the state-space equations into their variational form. This process introduces an inherent freedom, similar to the gauge freedom in electromagnetism. We then utilize the gauge function to reduce the numerical integration error. We show that by choosing an appropriate gauge function the numerical integration error dramatically decreases and one can achieve much better accuracy compared to the standard state variables for a given time-step. Moreover, we derive general expressions yielding the optimal gauge functions given a Newtonian one-degree-of-freedom ODE. For the n degrees-of-freedom case we describe a MATLAB code capable of finding the optimal gauge functions and integrating the given system using the gauge-optimized integration algorithm. In all of our illustrating examples, the gauge-optimized integration outperforms the integration using standard state variables by a few orders.

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