2,954 research outputs found
On the pathwise approximation of stochastic differential equations
We consider one-step methods for integrating stochastic differential
equations and prove pathwise convergence using ideas from rough path theory. In
contrast to alternative theories of pathwise convergence, no knowledge is
required of convergence in pth mean and the analysis starts from a pathwise
bound on the sum of the truncation errors. We show how the theory is applied to
the Euler-Maruyama method with fixed and adaptive time-stepping strategies. The
assumption on the truncation errors suggests an error-control strategy and we
implement this as an adaptive time-stepping Euler-Maruyama method using bounded
diffusions. We prove the adaptive method converges and show some computational
experiments.Comment: 21 page
Algorithms and Data Structures for Multi-Adaptive Time-Stepping
Multi-adaptive Galerkin methods are extensions of the standard continuous and
discontinuous Galerkin methods for the numerical solution of initial value
problems for ordinary or partial differential equations. In particular, the
multi-adaptive methods allow individual and adaptive time steps to be used for
different components or in different regions of space. We present algorithms
for efficient multi-adaptive time-stepping, including the recursive
construction of time slabs and adaptive time step selection. We also present
data structures for efficient storage and interpolation of the multi-adaptive
solution. The efficiency of the proposed algorithms and data structures is
demonstrated for a series of benchmark problems.Comment: ACM Transactions on Mathematical Software 35(3), 24 pages (2008
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is
solved numerically by using the finite difference method in combination with a
convex splitting technique of the energy functional. For the non-stochastic
case, we develop an unconditionally energy stable difference scheme which is
proved to be uniquely solvable. For the stochastic case, by adopting the same
splitting of the energy functional, we construct a similar and uniquely
solvable difference scheme with the discretized stochastic term. The resulted
schemes are nonlinear and solved by Newton iteration. For the long time
simulation, an adaptive time stepping strategy is developed based on both
first- and second-order derivatives of the energy. Numerical experiments are
carried out to verify the energy stability, the efficiency of the adaptive time
stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic
MATEX: A Distributed Framework for Transient Simulation of Power Distribution Networks
We proposed MATEX, a distributed framework for transient simulation of power
distribution networks (PDNs). MATEX utilizes matrix exponential kernel with
Krylov subspace approximations to solve differential equations of linear
circuit. First, the whole simulation task is divided into subtasks based on
decompositions of current sources, in order to reduce the computational
overheads. Then these subtasks are distributed to different computing nodes and
processed in parallel. Within each node, after the matrix factorization at the
beginning of simulation, the adaptive time stepping solver is performed without
extra matrix re-factorizations. MATEX overcomes the stiff-ness hinder of
previous matrix exponential-based circuit simulator by rational Krylov subspace
method, which leads to larger step sizes with smaller dimensions of Krylov
subspace bases and highly accelerates the whole computation. MATEX outperforms
both traditional fixed and adaptive time stepping methods, e.g., achieving
around 13X over the trapezoidal framework with fixed time step for the IBM
power grid benchmarks.Comment: ACM/IEEE DAC 2014. arXiv admin note: substantial text overlap with
arXiv:1505.0669
High-order adaptive time stepping for vesicle suspensions with viscosity contrast
We construct a high-order adaptive time stepping scheme for vesicle
suspensions with viscosity contrast. The high-order accuracy is achieved using
a spectral deferred correction (SDC) method, and adaptivity is achieved by
estimating the local truncation error with the numerical error of physically
constant values. Numerical examples demonstrate that our method can handle
suspensions with vesicles that are tumbling, tank-treading, or both. Moreover,
we demonstrate that a user-prescribed tolerance can be automatically achieved
for simulations with long time horizons
A locally adaptive time-stepping algorithm for\ud petroleum reservoir simulations
An algorithm for locally adapting the step-size for large scale finite volume simulations of multi-phase flow in petroleum reservoirs is suggested which allows for an “all-in-one” implicit calculation of behaviour over a very large time scale. Some numerical results for simple two-phase flow in one space dimension illustrate the promise of the algorithm, which has also been applied to very simple 3D cases. A description of the algorithm is presented here along with early results. Further development of the technique is hoped to facilitate useful scaling properties
Adaptive time-stepping for incompressible flow. Part II: Navier-Stokes equations
We outline a new class of robust and efficient methods for solving the Navier- Stokes equations. We describe a general solution strategy that has two basic building blocks: an implicit time integrator using a stabilized trapezoid rule with an explicit Adams-Bashforth method for error control, and a robust Krylov subspace solver for the spatially discretized system. We present numerical experiments illustrating the potential of our approach. © 2010 Society for Industrial and Applied Mathematics
Adaptive time-stepping for incompressible flow part I: scalar advection-diffusion
Even the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order trapezoid rule using an explicit Adams–Bashforth method for error control is assessed in this work. This combination is particularly well suited to long time integration of advection-dominated problems. Herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations: specifically diffusion problems with rough initial data; and general advection-diffusion problems with different physical time scales governing the system evolution
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