Textbook in PDF format Dedication Contents List of Figures List of Tables Preface to the First Edition Preface to the Second Edition What is geometric numerical integration? Simple harmonic oscillator Some elementary numerical methods Simple mathematical pendulum Classical paradigm of numerical integration Adjoint method, symmetric method Stability Stiffness Towards a new paradigm: Geometric numerical integration Symplectic integration Crash course on Hamiltonian dynamics: Hamiltonian systems Symplectic integrators of first and second-order Illustration: the Kepler problem Error growth Efficiency plots What is to be treated in this book (and what is not)? Exercises Taylor series methods Introduction General formulation Collocation methods Partitioned Runge–Kutta methods Runge–Kutta–Nyström methods Preservation of invariants in Runge–Kutta schemes Numerical examples Exercises Splitting and composition methods Lie–Trotter and Strang splitting Composition of Strang maps Composition of Lie-Trotter maps Splitting methods Composition methods Negative coefficients Stability RKN splitting methods Methods with commutators Near-integrable systems Processing Splitting methods for non-autonomous systems Some efficient splitting and composition methods General (non-processed) methods Processed methods Examples of application Exercises Other types of geometric numerical integrators Crash course on Hamiltonian dynamics II: Canonical transformations and generating functions Symplectic integrators based on generating functions Crash course on Hamiltonian dynamics III: Hamilton’s principle and Lagrangian formulation An introduction to variational integrators Volume-preserving methods Lie group methods Linear matrix equations: Magnus expansion Numerical schemes based on the Magnus expansion Nonlinear matrix equations in Lie groups Crouch–Grossman methods Positivity-preserving methods Linear combinations of splitting methods Exercises Introduction Examples Modified equations Modified equations of splitting methods Some physical insight Estimates over long-time intervals Exercises Introduction Splitting methods for the time-dependent Schrödinger equation Space discretization Splitting methods for time propagation Splitting methods for non-linear Schrödinger equations Splitting methods for parabolic evolution equations Splitting and composition methods with complex coefficients Compositions Palindromic and symmetric-conjugate splitting methods Alternating-conjugate splitting methods Exercises The gravitational N-body problem Hamiltonian Monte Carlo Hamiltonian simulation in quantum computing Exercises Differential equations, vector fields and flows Numerical integrators and series of differential operators Lie algebras The Lie algebra of Hamiltonian functions Structure constants Lie groups Free Lie algebras The Baker–Campbell–Hausdorff formul Bibliography Index