5 research outputs found
Adaptive Energy Preserving Methods for Partial Differential Equations
A method for constructing first integral preserving numerical schemes for
time-dependent partial differential equations on non-uniform grids is
presented. The method can be used with both finite difference and partition of
unity approaches, thereby also including finite element approaches. The schemes
are then extended to accommodate -, - and -adaptivity. The method is
applied to the Korteweg-de Vries equation and the Sine-Gordon equation and
results from numerical experiments are presented.Comment: 27 pages; some changes to notation and figure
Discrete gradients in short-range molecular dynamics simulations
Discrete gradients (DG) or more exactly discrete gradient methods are time
integration schemes that are custom-built to preserve first integrals or
Lyapunov functions of a given ordinary differential equation (ODE). In
conservative molecular dynamics (MD) simulations, the energy of the system is
constant and therefore a first integral of motion. Hence, discrete gradient
methods seem to be a natural choice as an integration scheme in conservative
molecular dynamics simulations
Order theory for discrete gradient methods
We present a subclass of the discrete gradient methods, which are integrators
designed to preserve invariants of ordinary differential equations. From a
formal series expansion of the methods, we derive conditions for arbitrarily
high order. We devote considerable space to the average vector field discrete
gradient, from which we get P-series methods in the general case, and B-series
methods for canonical Hamiltonian systems. Higher order schemes are presented
and applied to the H\'enon-Heiles system and a Lotka-Volterra system.Comment: 45 pages, 5 figure
エネルギー関数を持つ発展方程式に対する幾何学的数値計算法
学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学