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Probabilistic Evolution Theory in Subintervals with Condensed Kronecker Product for Explicit Autonomous ODEs: Classical Two-Particle Problem as an Example

Last modified: 2020-01-15

#### Abstract

Probabilistic evolution theory (PREVTH) is a powerful tool for the solution of explicit ODEs with multinomial right-hand side functions. PREVTH in subintervals is important for obtaining convergence. Condensed Kronecker product facilitates the use of smaller vectors and matrices by getting rid of unnecessary flexibilities. All these developments have made PREVTH into a powerful and easy-to-use method. In this work, we use PREVTH for the solution of classical two-particle problem. Our purpose is to make observations about the convergence by manipulating subinterval length and order of truncation. Our main observation is that the method forms very large and very small numbers during the calculation. Multiplication of very large and very small numbers accumulates error quickly and causes loss of significant digits. Although approaches such as arbitrary-precision or exact-precision arithmetic may help solve the problem of error accumulation, they come at a high computational cost.