Adaptive Time-stepping Method Research Articles

A deterministic scheme for Smoluchowski's coagulation equation based on binary grid refinement

We present a deterministic scheme for the discrete Smoluchowski's coagulation equation based on a binary grid refinement. Starting from the binary grid Ω0={1,2,4,8,16,. . .}, we first introduce an appropriate grid refinement by adding at each level 2l grid points in every binary subsection of the grid Ωl. In a next step we derive an approximate equation for the dynamic behavior on each level Ωl based on a piecewise constant approximation of the right hand side of Smoluchowski's equation. Numerical results show that the computational effort can be drastically decreased compared to the corresponding complete integer grid. When considering unbounded kernels in Smoluchowski's equation we use an adaptive time step method to overcome numerical instabilities which may occur at the tails of the density function.

The Effective Stability of Adaptive Timestepping ODE Solvers

We consider the behavior of certain adaptive timestepping methods, based upon embedded explicit Runge--Kutta pairs, when applied to dissipative ODEs. It has been observed numerically that the standard local error controls can impart desirable stability properties, but this has been rigorously verified only for very special, low-order, Runge--Kutta pairs. The rooted-tree expansion of a certain quadratic form, central to the stability theory of Runge--Kutta methods, is derived. This, together with key assumptions on the sequence of accepted time-steps and the local error estimate, provides a general explanation for the observed stability of such algorithms on dissipative problems. Under these assumptions, which are expected to hold for typical numerical trajectories, two different results are proved. First, for a large class of embedded Runge--Kutta pairs of order $(1,2)$, controlled on an error-per-unit-step basis, all such numerical trajectories will eventually enter a particular bounded set. This occurs for sufficiently small tolerances independent of the initial conditions. Second, for pairs of arbitrary orders $(p-1,p)$, operating under either error-per-step or error-per-unit-step control, similar results are obtained when an additional structural assumption (that should be valid for many cases of interest) is imposed on the dissipative vector field. Numerical results support both the analysis and the assumptions made.

Adaptive weak approximation of stochastic differential equations

AbstractAdaptive time‐stepping methods based on the Monte Carlo Euler method for weak approximation of Itô stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading‐order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps. © 2001 John Wiley & Sons, Inc.

Additional efficient computation of branched nerve equations: adaptive time step and ideal voltage clamp.

Various improvements are described for the simulation of biophysically and anatomically detailed compartmental models of single neurons and networks of neurons. These include adaptive time-step integration and a reordering of the circuit matrix to allow ideal voltage clamp of arbitrary nodes. We demonstrate how the adaptive time-step method can give equivalent accuracy as a fixed time-step method for typical current clamp simulation protocols, with about a 2.5 reduction in runtime. The ideal voltage clamp method is shown to be more stable than the nonideal case, in particular when used with the adaptive time-step method. Simulation results are presented using the Surf-Hippo Neuron Simulation System, a public domain object-oriented simulator written in Lisp.

An evaluation of temporally adaptive transformation approaches for solving Richards' equation

Developing robust and efficient numerical solution methods for Richards' equation (RE) continues to be a challenge for certain problems. We consider such a problem here: infiltration into unsaturated porous media initially at static conditions for uniform and non-uniform pore size media. For ponded boundary conditions, a sharp infiltration front results, which propagates through the media. We evaluate the resultant solution method for robustness and efficiency using combinations of variable transformation and adaptive time-stepping methods. Transformation methods introduce a change of variable that results in a smoother solution, which is more amenable to efficient numerical solution. We use adaptive time-stepping methods to adjust the time-step size, and in some cases the order of the solution method, to meet a constraint on nonlinear solution convergence properties or a solution error criterion. Results for three test problems showed that adaptive time-stepping methods provided robust solutions; in most cases transforming the dependent variable led to more efficient solutions than untransformed approaches, especially as the pore-size uniformity increased; and the higher-order adaptive time integration method was robust and the most efficient method evaluated.

An advanced algorithm for solving partial differential equation in cardiac conduction.

An advanced integration method for solving reaction-diffusion-type equations for cardiac conduction is suggested. Operator splitting and adaptive time step methods were used in this method, which can significantly speed up integration while preserving accuracy.

Computational micro- and macroscopic models of contact and friction: formulation, approach and applications

This paper is dedicated to new asperity-based constitutive models of contact interfaces. These models have been obtained through a combination of finite element analysis of surface asperities and statistical homogenization techniques, to predict macroscopic, phenomenological behavior of the interface. This new approach has generalized the existing asperity-based models of contact and friction by considering realistic, complex shapes and mechanical properties of surface asperities, as opposed to previous simplified analytical solutions. This has been achieved by application, at the stage of asperity modeling, of the finite element method, which takes into account arbitrary shapes of asperities, non-linear material properties, molecular-range adhesion forces, and sliding resistance on the contact surface. The h–p adaptive mesh refinement techniques, adaptive timestepping and other adaptive methods are used to assure high accuracy of the solution. The result of this development is a new family of constitutive interface models, consistent with surface micromechanics and applicable to studies of static and dynamic friction phenomena. They are also extendible to calculation of thermal or electrical resistances, wear modeling, and other applications. This paper presents the theoretical formulation, numerical methodology and sample models of contact, adhesion and friction obtained through these homogenization techniques.

A Second-Derivative-Based Adaptive Time-Step Method for Spatial Kinetics Calculations

In solving few-group neutron kinetic equations in multidimensions, one must select time step sizes as a function of time such that the temporal truncation error introduced by the discrete time deri...

Adaptive finite element methods for continuum damage modeling

The paper presents an application of adaptive finite element methods to the modeling of low-cycle continuum damage and life prediction of high-temperature components. The major objective is to provide automated and accurate modeling of damaged zones through adaptive mesh refinement and adaptive time-stepping methods. The damage modeling methodology is implemented in a usual way by embedding damage evolution in the transient nonlinear solution of elasto-viscoplastic deformation problems. This nonlinear boundary-value problem is discretized by adaptive finite element methods. The automated h-adaptive mesh refinements are driven by error indicators, based on selected principal variables in the problem (stresses, nonelastic strains, damage, etc.). In the time domain, adaptive time-stepping is used, combined with a predictor-corrector time marching algorithm. The time step selection is controlled by required time accuracy. In order to take into account strong temperature dependency of material parameters, the nonlinear structural solution is coupled with thermal analyses (one-way coupling). Several test examples illustrate the importance and benefits of adaptive mesh refinements in accurate prediction of damage levels and failure time.

High strain-rate localization and failure of crystalline materials

A theoretical and a computational model are introduced to study the micromechanical high strain-rate failure mechanisms of shear-strain localization in monocrystalline fcc structures. A theoretical framework for a constitutive model for the dynamic finite plastic deformation of rate-dependent single fcc crystals is developed. The micromechanics of plastic flow are based on a high strain-rate single crystal plasticity model and a visco-plastic power law. The single crystal is subjected to far-field dynamic tensile strain-rates ranging from 100/s to 2000/s. An explicit finite-element model is introduced for the integration of the numerically stiff visco-plastic constitutive relations. The total deformation rate tensor is obtained by the central difference explicit integration of the equations of motion. The plastic deformation rate tensor is obtained from the solution of an initial-value nonlinear problem for the resolved shear stresses. In time intervals when the differential equations for the resolved shear stresses are not numerically stiff, the initial-value problem is integrated by the explicit fifth-order Runge-Kutta adaptive time-step method. In time domains where the propagated error grows and the time-step must be restricted due to stability requirements, which is an indication of numerical stiffness, the initial-value problem is integrated by an A- stable method . To correctly differentiate time-step reductions due to stability, from time-step reductions due to accuracy, a stiffness ratio is defined. The present analysis corroborates experimental observations that high strain-rate shear-strain localization, in rate-dependent crystals, is a function of thermal and geometrical softening, overall strain-rates, strain hardening, strain-rate hardening, and strain-rate sensitivity.