## Degree

Doctor of Philosophy (PhD)

## Department

Mathematics

## Document Type

Dissertation

## Abstract

The goal of this dissertation is to apply the concept of Lie generators for linear semigroups induced by nonlinear flows, originally developed by J. R. Dorroh and J. W. Neuberger in the 1990’s [15], to approximate solutions of initial value problems like

x′(t) = F(x(t)), x(0) = x_{0}, (1)

where F = (F_{1},··· ,F_{N}), and F_{i} : R^{N} ⊃ Ω -> R^{N}. The method, sometimes referred to as ``Bernard Koopman’s Global Linearization Method,” traces its origins back to the works of Sophus Lie in the 1890’s [30], Gerhard Kowalewski in 1931 [28], and Wolfgang Gr¨ obner in the 1960’s [21]. They recognized the advantages of analysing the nonlinear problem (1) in terms of the linear Koopman-Lie semigroup e^{tK} g(x_{0}) = g(x(t)), where

Kg(x) = (F_{1}(x) δ/ δx_{1} +F_{2}(x) δ/ δx_{2} +··· +F_{N}(x) δ/ δx_{N} ) g(x) =(K_{1}+···+K_{N})g(x),

and g : R^{N} -> R is some observation of the solution t -> x(t). Our approach differs from the classical one in that we do not concentrate on the Lie series

e^{tK} = I +tK+ t^{2} /2! K^{2} +··· ,

but on product formulas like the well-known Lie-Trotter product formula

g(x(t)) = e^{tK} g(x) = e^{tK1+tK2+...+tKN} g(x) = lim_{n -> ∞} (e^{t/n KN}···e^{t/n K1})^{n}g(x),

which converges of order t^{2} /n . In this work, we investigate higher order splitting methods with order t^{p+1} /n^{p} for p ≥ 1 for N-dimensional splittings K = K_{1} +K_{2} + ··· + K_{N}, both theoretically and numerically.

## Date

7-12-2024

## Recommended Citation

Banjara, Arun, "Higher Order Operator Splitting Schemes with Complex Coefficients and Applications" (2024). *LSU Doctoral Dissertations*. 6529.

https://repository.lsu.edu/gradschool_dissertations/6529

## Committee Chair

Neubrander, Frank

## DOI

10.31390/gradschool_dissertations.6529