Adaptive Time-stepping Scheme Research Articles

Adaptive [formula omitted]-methods for pricing American options

We develop adaptive θ -methods for solving the Black–Scholes PDE for American options. By adding a small, continuous term, the Black–Scholes PDE becomes an advection–diffusion-reaction equation on a fixed spatial domain. Standard implementation of θ -methods would require a Newton-type iterative procedure at each time step thereby increasing the computational complexity of the methods. Our linearly implicit approach avoids such complications. We establish a general framework under which θ -methods satisfy a discrete version of the positivity constraint characteristic of American options, and numerically demonstrate the sensitivity of the constraint. The positivity results are established for the single-asset and independent two-asset models. In addition, we have incorporated and analyzed an adaptive time-step control strategy to increase the computational efficiency. Numerical experiments are presented for one- and two-asset American options, using adaptive exponential splitting for two-asset problems. The approach is compared with an iterative solution of the two-asset problem in terms of computational efficiency.

An adaptive Euler-Maruyama scheme for SDEs: convergence and stability

The understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded, then explicit fixed time step approximations, such as the Euler–Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself.

Adaptive time-stepping and computational stability

We investigate the effects of adaptive time-stepping and other algorithmic strategies on the computational stability of ODE codes. We show that carefully designed adaptive algorithms have a most significant impact on computational stability and reliability. A series of computational experiments with the standard implementation of D ASSL and a modified version, including stepsize control based on digital filters, is used to demonstrate that relatively small algorithmic changes are able to extract a vastly better computational stability at no extra expense. The inherent performance and stability of D ASSL are therefore much greater than the standard implementation seems to suggest.

Mean-square stability properties of an adaptive time-stepping SDE solver

We consider stability properties of a class of adaptive time-stepping schemes based upon the Milstein method for stochastic differential equations with a single scalar forcing. In particular, we focus upon mean-square stability for a class of linear test problems with multiplicative noise. We demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls, one for the drift term and one for the diffusion.

An adaptive hybrid time‐stepping scheme for highly non‐linear strongly coupled problems

AbstractThis paper deals with the design and implementation of an adaptive hybrid scheme for the solution of highly non‐linear, strongly coupled problems. The term ‘hybrid’ refers to a composite time stepping scheme where a controller decides whether a monolithic scheme or a fractional step (splitting) scheme is appropriate for a given time step. The criteria are based on accuracy and efficiency. The key contribution of this paper is the development of a framework for incorporating error criteria for stepsize selection and a mechanism for choosing from splitting or monolithic possibilities. The resulting framework is applied to silylation, a highly non‐linear, strongly coupled problem of solvent diffusion and reaction in deforming polymers. Numerical examples show the efficacy of our new hybrid scheme on both two‐ and three‐dimensional silylation simulations in the context of microlithography. Copyright © 2005 John Wiley & Sons, Ltd.

A multigrid finite element solver for the Cahn–Hilliard equation

A multigrid finite element solver for the Cahn–Hilliard equation is presented that has mesh-independent convergence rates for any time-step size, including in the important limit ϵ → 0 which is examined via numerical examples. Numerics are performed for a number of test problems which show that the features of the Cahn–Hilliard equation (minimising interface measure, Lyapunov energy functional etc.) are preserved. We also explore the use of this solver in conjunction with adaptive time-stepping and adaptive mesh strategies.

An adaptive time step solution for raster-based storage cell modelling of floodplain inundation

Since 1962 storage cell codes have been developed to simulate flow on fluvial and coastal floodplains. These models treat the floodplain as a series of discrete storage cells, with the flow between cells calculated explicitly using some analytical flow formulae such as the Manning equation. Recently these codes have been reconfigured to use regular Cartesian grids to make full use of widely available high resolution data captured from remote sensing platforms and stored in a raster GIS format. Such raster-based storage cell codes have many of the advantages over full two-dimensional depth averaged schemes but without the computational cost, however their typical implementation results in a number of fundamental limitations. These include an inability to develop solutions that are independent of time step or grid size, and an unrealistic lack of sensitivity to floodplain friction. In this paper, we propose a new solution to these problems based on an optimal adaptive time step determined using the Courant–Freidrichs–Levy condition for model stability. Comparison of this new adaptive time step scheme to analytical solutions of wave propagation on flat and sloping planar surfaces shows considerable improvement over a standard raster storage cell model. Moreover, the new scheme is shown to yield results that are independent of grid size or choice of initial time step and which show an intuitively correct sensitivity to floodplain friction over spatially complex topography.

Adaptive time-stepping analysis of nonlinear microwave heating problems

This paper presents a three-dimensional computational model for nonlinear microwave heating problems. In order to reduce the computational costs, a new finite element method adaptive time-stepping scheme is proposed. Numerical results are compared with experimental measurements on a problem of dielectric heating of water.

Pure ice sublimation within vials in a laboratory lyophiliser; comparison of theory with experiment

A two-dimensional axisymmetric, unsteady state model was developed to simulate the coupled heat and mass transfer process that occurs during the sublimation of pure ice from within vials in a laboratory-scale lyophiliser. The theoretical model was solved numerically using an explicit finite difference method. The modified Euler method was used to discretise in time and an adaptive time step strategy was employed to ensure stability. Novel features of the model described here include a consideration of the varying, pressure dependent, insulating effect of the air gap under the curved base of a typical lyophilisation vial and the determination of the effects of radiative heating. The theoretical model was capable of predicting and describing the observed motion of the ice sublimation front, the ice temperature profile, the drying time and the influence of the total pressure. The model also gave a theoretical prediction of the improvement in drying rate that might be achieved by adding an extra energy source. The work provides insights into the understanding of the heat and mass transfer process during the primary drying stage of lyophilisation that can be applied to the optimisation of freeze-drying cycle design.

Efficient and accurate time adaptive multigrid simulations of droplet spreading

AbstractAn efficient full approximation storage (FAS) Multigrid algorithm is used to solve a range of droplet spreading flows modelled as a coupled set of non‐linear lubrication equations. The algorithm is fully implicit and has embedded within it an adaptive time‐stepping scheme that enables the same to be optimized in a controlled manner subject to a specific error tolerance. The method is first validated against a range of analytical and existing numerical predictions commensurate with droplet spreading and then used to simulate a series of new, three‐dimensional flows consisting of droplet motion on substrates containing topographic and wetting heterogeneities. The latter are of particular interest and reveal how droplets can be made to spread preferentially on substrates owing to an interplay between different topographic and surface wetting characteristics. Copyright © 2004 John Wiley & Sons, Ltd.

A fully coupled physically-based spatially-distributed model for evaluating surface/subsurface flow

A physically-based, spatially-distributed model is presented for simulation of surface/subsurface flow and the interactions between these domains. The model is designed for practical application to a wide variety of hydrologic evaluations, at various scales of simulation. The system is represented by the three-dimensional saturated–unsaturated flow equation for the subsurface, coupled with the diffusion wave equation for areal overland flow, both of which are coupled with the diffusion wave equation for flow through a network of streams and channels, including hydraulic structures. Ground surface unevenness at the grid scale is incorporated via the concept of detention storage, and thick vegetation or urban features are included via an obstruction storage exclusion term. Evapotranspiration from the surface and subsurface are modeled using land cover and climatic factors to define the complete water budget using a physically-based formulation. The system of equations is discretized using a fully implicit procedure, with the Newton–Raphson method to handle non-linearities efficiently. Robustness, stability and accuracy of solution are obtained for a wide variety of cases including dry systems and large surface/subsurface interaction fluxes. Adaptive time-stepping schemes and under-relaxation formulas further alleviate the computational burden. Verification and application examples demonstrate the need for a rigorous, fully-coupled solution to the set of equations, for complete hydrologic-cycle analysis.

Reliable finite element analysis of ship structural components

Reliable and accurate determination of the static and dynamic response quantities is of considerable importance for economical and safe design of ship hull structures and offshore platforms. The objective of this paper is to present efficient methodologies that can be employed to obtain reliable finite element solutions for static and dynamic analysis of ship structural components. An a posteriori error estimator and h-adaptive refinement strategy is proposed for static and dynamic analysis of stiffened plate/shell panels. The equal subdivision or bisection method is made use of and local refinements are carried out only in the elements requiring refinements. An adaptive time-stepping scheme is proposed for transient dynamic analysis of stiffened plate/shell panels based on the time error estimates. Numerical studies on the static and dynamic analysis of ship hull structural components have been conducted to demonstrate the performance of the error estimator and the adaptive refinement strategy. It is observed that the proposed adaptive strategy in association with the error estimator is helpful in arriving at optimal mesh and time steps for the specified accuracy. It is recommended that the proposed methodologies can be used in obtaining reliable finite element solutions for design of ship structural components subjected to static and dynamic loads.

An effective algorithm for simulating diffusion-driven aggregation

In many fields of physics and chemistry growth phenomena driven by mutual particle collisions (coagulation) are of great interest. In this paper an effective N-particle-method for simulating the coagulation due to Brownian motion at very low number densities is presented. Under such conditions the mean particle distance exceeds the typical aggregate diameter by orders of magnitude and a collision will be a very rare event. We derive a criterion which allows an efficient detection of candidates for imminent collisions. The N-particle method is based on an adaptive time step scheme respecting for the individual dynamical states of the aggregates. The numerical cost of the algorithm scales with the particle number approximately as N logN . In order to minimize the influence of the decreasing number of particles within the simulation box a new rescaling method is used throughout the aggregation process.

Additive splitting methods for elliptic-parabolic problems

Three finite-difference splitting schemes are proposed for numerical solution of the nonlinear 3D parabolic-elliptic problem. Adaptive front-tracking and time-stepping strategies are included into the algorithms. Parallelization of the algorithms is done using the domain decomposition method. The 1D decomposition of the computational domain is used in order to obtain the optimal computational load balancing among processors and to minimize the frequency of data communications. A redistribution of the computational domain among processors is done dynamically during computations.

A time-adaptive space-time finite element method for incompressible Lagrangian flows with free surfaces: computational issues

In order to further enhance the performance of the space-time Galerkin/least-squares method for solving incompressible Navier–Stokes problems involving free surfaces, issues related to the solution strategy and time adaptivity are addressed . Due to the a priori unknown boundary positions, a nonlinear system of equations normally arises at each space-time slab, which is solved by the Newton–Raphson approach. In addition, a linear system of equations for velocity and pressure that provides an alternative approach to the solution of the problem is also derived in this paper. This linear solution scheme can significantly reduce the computational costs in terms of computer CPU time and memory requirements without the sacrifice of the solution accuracy if the time-step size is sufficiently small. Furthermore, the possibility of adaptively adjusting time-step size is fully exploited. By choosing the volume loss rate as an error indicator, a simple adaptive time-stepping scheme is presented. Finally several numerical examples are provided to assess the performances of the proposed schemes.

FLUX-BASED FINITE-VOLUME FORMULATIONS AND ADAPTIVE TIME-STEPPING STRATEGIES FOR MODELING OF REENTRY THERMAL PROTECTION SYSTEMS

We describe effective thermal modeling and analysis formulations of reusable thermal protection systems (TPS) used in reentry space vehicles employing flux-based finit-volume representations in conjunction with adaptive time-stepping strategies for accurately tracking the nonlinear transient thermal response of such configurations. The flux-based finite-volume representations are developed following the spirit of a Lax-Wendroff/finite-volume-type formulation. In conjunction with a trapezoidal γ T family of algorithms and adaptive automated time-stepping strategies that permit a good global error control, the representations offer an improved physical interpretation and mathematical representation of the thermal models with several computationally convenient and attractive features. The adaptive time stepping provides use of an optimal number of time steps in a single analysis and is effective in conjunction with the present physically improved flux-based finite-volume representations for the thermal analysis of large TPS models and the associated structural configurations.

Use of dynamically adaptive grid techniques for the solution of electrochemical kinetic equations: Part 3. An adaptive moving grid—adaptive time step strategy for problems with discontinuous boundary conditions at the electrodes

The finite-difference adaptive moving grid strategy suggested in Parts 1 and 2 for the solution of electrochemical kinetic one-dimensional partial differential equations has been further extended to allow temporal grid adaptation as well as time-stepping across discontinuities in boundary conditions. This extension aims at the development of a general simulation algorithm, capable of a largely automatic solution of a variety of kinetic problems. The validity of the strategy developed has been tested on three examples of the modelling of electrochemical transients: square-wave-controlled potential transient in pure diffusion conditions, potential-step transient for the catalytic electrode reaction mechanism with a fast homogeneous reaction and linear potential scan voltammetric transient for an electrode reaction accompanied by a fast follow-up homogeneous dimerization reaction.

An adaptive time-stepping scheme for the viscoplasticity theory based on overstress

A finite element time-integration method for the viscoplasticity theory based on overstress with variable time steps is developed. The criterion for time-step change is based on the truncation error of multistep methods modified by the magnitude of the deviatoric strain tensor. Time-step changes are affected by an empirical formula which captures the characteristics of the viscoplasticity theory based on overstress. The adaptive scheme is incorporated in the forward gradient (one-step method) and two-step methods developed previously. Results for uniaxial loading and for an internally pressurized cylinder under plane strain indicate that the respective adaptive methods are more accurate and more economical than the forward gradient and the two-step methods.

Moisture transport in a soil-plant system: a mathematical model and finite element analysis

A mathematical model and approximate analysis using finite elements are developed to simulate the transport of moisture from a soil medium through a small seedling or plant to the atmosphere. An intrinsic part of the mathematical model is analysis of the characteristic diffusion rates of different components in the soil-plant system. This leads to consideration of the non-linear coupling of the component soil and plant regimes and effective numerical solution of the problem using a Galerkin semi-discrete finite element method: Details concerning the mathematical model and approximate analysis are described. Numerical experiments are conducted to examine the performance of the model. Relaxation and iterative acceleration techniques and an adaptive timestepping strategy are devised to improve the solution algorithms. Comparisons of model predictions with results of laboratory and field experiments indicate that the model provides useful information on plant-water relations.