918 research outputs found
Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations
We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problem
Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise
A fully discrete approximation of the semi-linear stochastic wave equation
driven by multiplicative noise is presented. A standard linear finite element
approximation is used in space and a stochastic trigonometric method for the
temporal approximation. This explicit time integrator allows for mean-square
error bounds independent of the space discretisation and thus do not suffer
from a step size restriction as in the often used St\"ormer-Verlet-leap-frog
scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift
of the expected value of the energy of the problem). Numerical experiments are
presented and confirm the theoretical results
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
We introduce a new class of integrators for stiff ODEs as well as SDEs. These
integrators are (i) {\it Multiscale}: they are based on flow averaging and so
do not fully resolve the fast variables and have a computational cost
determined by slow variables (ii) {\it Versatile}: the method is based on
averaging the flows of the given dynamical system (which may have hidden slow
and fast processes) instead of averaging the instantaneous drift of assumed
separated slow and fast processes. This bypasses the need for identifying
explicitly (or numerically) the slow or fast variables (iii) {\it
Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time
scale can be used as a black box and easily turned into one of the integrators
in this paper by turning the large coefficients on over a microscopic timescale
and off during a mesoscopic timescale (iv) {\it Convergent over two scales}:
strongly over slow processes and in the sense of measures over fast ones. We
introduce the related notion of two-scale flow convergence and analyze the
convergence of these integrators under the induced topology (v) {\it Structure
preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be
made to be symplectic, time-reversible, and symmetry preserving (symmetries are
group actions that leave the system invariant) in all variables. They are
explicit and applicable to arbitrary stiff potentials (that need not be
quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy
and stability over four orders of magnitude of time scales. For stiff Langevin
equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs
reversible, quasi-symplectic on all variables and conformally symplectic with
isotropic friction.Comment: 69 pages, 21 figure
Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics
This paper presents an overview of physical ideas and mathematical methods
for implementing non-smooth and discontinuous substitutions in dynamical
systems. General purpose of such substitutions is to bring the differential
equations of motion to the form, which is convenient for further use of
analytical and numerical methods of analyses. Three different types of
nonsmooth transformations are discussed as follows: positional coordinate
transformation, state variables transformation, and temporal transformations.
Illustrating examples are provided.Comment: 15 figure
From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials
We present a multiscale integrator for Hamiltonian systems with slowly
varying quadratic stiff potentials that uses coarse timesteps (analogous to
what the impulse method uses for constant quadratic stiff potentials). This
method is based on the highly-non-trivial introduction of two efficient
symplectic schemes for exponentiations of matrices that only require O(n)
matrix multiplications operations at each coarse time step for a preset small
number n. The proposed integrator is shown to be (i) uniformly convergent on
positions; (ii) symplectic in both slow and fast variables; (iii) well adapted
to high dimensional systems. Our framework also provides a general method for
iteratively exponentiating a slowly varying sequence of (possibly high
dimensional) matrices in an efficient way
Fourier spectral methods for fractional-in-space reaction-diffusion equations
Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional AllenāCahn, FitzHughāNagumo and GrayāScott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator
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