Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators

Abstract

We describe a package realized in the Julia programming language which performs symbolic manipulations applied to nonlinear evolution equations, their flows, and commutators of such objects. This tool was employed to perform contrived computations arising in the analysis of the local error of operator splitting methods. It enabled the proof of the convergence of the basic method and of the asymptotical correctness of a defect-based error estimator. The performance of our package is illustrated on several examples. Problem settingWe are interested in the solution to nonlinear evolution equationson a Banach space X, where A and B are general nonlinear, unbounded operators defined on a subset D ⊂ X, the solution is denoted by E H (t, u 0 ), and analogously for the two sub-flows associated with A and B. The structure of the vector fields often suggests to employ additive splitting methods to separately propagate the two subproblems defined by A and B,