# Higher Order Operator Splitting Schemes with Complex Coefficients and Applications

## Degree

Doctor of Philosophy (PhD)

Mathematics

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) = x0, (1)

where F = (F1,··· ,FN), and Fi : RN ⊃ Ω -> RN. 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 etK g(x0) = g(x(t)), where

Kg(x) = (F1(x) δ/ δx1 +F2(x) δ/ δx2 +··· +FN(x) δ/ δxN ) g(x) =(K1+···+KN)g(x),

and g : RN -> 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

etK = I +tK+ t2 /2! K2 +··· ,

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

g(x(t)) = etK g(x) = etK1+tK2+...+tKN g(x) = limn -> ∞ (et/n KN···et/n K1)ng(x),

which converges of order t2 /n . In this work, we investigate higher order splitting methods with order tp+1 /np for p ≥ 1 for N-dimensional splittings K = K1 +K2 + ··· + KN, both theoretically and numerically.

7-12-2024

## Committee Chair

Neubrander, Frank