Time-stepping Process Research Articles

Generalized Finite Integration Method with Laplace transform for European option pricing under Black–Scholes and Heston models

In this paper, we combine a recently developed Generalized Finite Integration Method (GFIM) with Laplace transform technique for pricing options under the Black Scholes model and Heston model respectively. Instead of using traditional time-stepping process, we first perform Laplace transform on the governing equation and boundary conditions to remove the temporal derivatives. The Generalized Finite Integration Method is then exploited to handle the spatial differential operators in the transformed space. From numerical Laplace inversion algorithm, we restore the required time-dependent option price. For verification, several option pricing models governed by one-dimensional Black–Scholes equation and two-dimensional extended Heston equation are constructed to demonstrate the efficiency and feasibility of the proposed approach.

On the application of the GS4-1 framework for fluid dynamics and adaptive time-stepping via a universal A-posteriori error estimator

PurposeThe purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.Design/methodology/approachThe proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.FindingsThe proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.Originality/valueTo the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

Quantitative modeling of residential building disaster recovery and effects of pre- and post-event policies

Understanding the process of community recovery impacted by various post-disaster decisions such as dynamic policies can effectively guide the recovery process and expedite recovery, thereby establishing a more resilient community. This paper proposes a methodology based on a multi-layer Monte Carlo simulation to model a two-stage recovery process for residential buildings: functional downtime due to delay and functional downtime due to repair. The delay portion of the model was modified based on the REDi framework and models the impeding factors that delay repairs such as post-disaster inspection, insurance claims, and building permits. Household income was examined to estimate the financing delay depending on different funding resources such as insurance and loans available to households at different income levels. The repair portion of the model followed the FEMA P-58 approach (which was originally for post-earthquake analysis) and was controlled by fragility functions. This study also investigates a series of policies to examine an illustrative example, namely the 2011 Joplin tornado. The residential recovery is modeled as a time-stepping process without any empirical data such that policies can be implemented pre-disaster (mitigation) and/or post-disaster. The ability to model hypothetical policy scenarios for residential recovery of a community will enable decision-makers to better understand collective community-wide impacts of their actions and policies, thereby improving community resilience planning.

Mathematical theory and simulations of thermoporoelasticity

In this paper we study the equations of semi-linear thermoporoelasticity. Starting point is the dimensionless formulation in van Duijn et al. (2019), which was obtained by a formal two-scale expansion. Nonlinearities in the equations arise through the fluid viscosity and the thermal conductivity, both may depend on temperature, and through the coupling in the heat convection by the Darcy discharge in the energy equation. The coupled system of equations involves as unknowns the skeleton displacement, Darcy discharge, fluid pressure and temperature. We treat the system in its incremental (i.e. time-discrete) form. We prove existence by applying a fundamental theorem of Brézis on pseudo-monotone operators. Moreover we show that the free energy of the system acts as a Lyapunov functional. This yields global stability in the time-stepping process. Our theoretical results are substantiated with two-dimensional numerical tests using a monolithic formulation. Temporal discretization is based on the backward Euler scheme and finite elements are employed for the spatial discretization. The semi-linear discrete system is solved with Newton’s method. In the proposed numerical examples, different source terms are employed and spatial mesh refinement studies show computational convergence.

A matrix-exponential decomposition based time-domain method for calculating the defect states of scalar waves in two-dimensional periodic structures

A time-domain method for calculating the defect states of scalar waves in two-dimensional (2D) periodic structures is proposed. In the time-stepping process of the proposed method, the column vector containing the spatially sampled field values is updated by multiplying it with an iteration matrix, which is written in a matrix-exponential form. The matrix-exponential is first computed by using the Suzuki's decomposition based technique of the fourth order, in which the Floquet–Bloch boundary conditions are incorporated. The obtained iteration matrix is then squared to enlarge the time-step that can be used in the time-stepping process (namely, the squaring technique), and the small nonzero elements in the iteration matrix is finally pruned to improve the sparse structure of the matrix (namely, the pruning technique). The numerical examples of the super-cell calculations for 2D defect-containing phononic crystal structures show that, the fourth order decomposition based technique for the matrix-exponential computation is much more efficient than the frequently used precise integration technique (PIT) if the PIT is of an order greater than 2. Although it is not unconditionally stable, the proposed time-domain method is particularly efficient for the super-cell calculations of the defect states in a 2D periodic structure containing a defect with a wave speed much higher than those of the background materials. For this kind of defect-containing structures, the time-stepping process can run stably for a sufficiently large number of the time-steps with a time-step much larger than the Courant–Friedrichs–Lewy (CFL) upper limit, and consequently the overall efficiency of the proposed time-domain method can be significantly higher than that of the conventional finite-difference time-domain (FDTD) method. Some physical interpretations on the properties of the band structures and the defect states of the calculated periodic structures are also presented.

Two dimensional fully nonlinear numerical wave tank based on the BEM

The development of a two dimensional numerical wave tank (NWT) with a rocker or piston type wavemaker based on the high order boundary element method (BEM) and mixed Eulerian-Lagrangian (MEL) is examined. The cauchy principle value (CPV) integral is calculated by a special Gauss type quadrature and a change of variable. In addition the explicit truncated Taylor expansion formula is employed in the time-stepping process. A modified double nodes method is assumed to tackle the corner problem, as well as the damping zone technique is used to absorb the propagation of the free surface wave at the end of the tank. A variety of waves are generated by the NWT, for example; a monochromatic wave, solitary wave and irregular wave. The results confirm the NWT model is efficient and stable.

An Adaptive Mesh Method in Transient Finite Element Analysis of Magnetic Field Using a Novel Error Estimator

A novel error estimator for adaptive mesh refinement in finite element analysis (FEA) of magnetic field using quadratic finite elements is presented. The method uses a novel heuristic a posteriori error estimator, which is easy to compute and simple to implement, as an indicator of the numerical errors of the computed solution. The proposed error estimator is the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> norm of the difference between the computed quadratic finite element solution and the interpolated linear solution. Throughout the time-stepping process for problems excited by periodic sinusoidal excitations, this error estimator can also be used to efficiently compute the numerical error at each time step and guide the adaptive mesh refinement in transient FEA. A multi-time-step adaptive mesh refinement method is also proposed in this paper for transient problems. The proposed method does not need to interpolate the solution from old mesh to a new adaptively refined mesh in transient FEA and hence there is no interpolation error. The effectiveness of the proposed error estimator is illustrated through several numerical examples being reported in this paper.

Concerning the cause of instability in time-stepping boundary element methods applied to the exterior acoustic problem

The boundary element method (BEM) in its simple form when solving the exterior acoustic problem in the frequency domain has difficulties at the frequencies of internal resonances of the closed structure. The corresponding time domain form of the exterior problem often exhibits instabilities in the time-stepping process. The link between these two features is investigated by relating the eigenvalues of the iterating matrix of the time-stepping process to damped frequencies. Numerical evidence from a problem with an analytic solution, and from a loudspeaker response are given. The suggested link implies that instability comes from numerical errors super-imposed on a more fundamental problem and may be best tackled through a time domain technique corresponding to the methods already available in the frequency domain.

Two-dimensional equilibrium morphological modelling of a tidal inlet: an entropy based approach

The management of tidal inlets requires the accurate prediction of equilibrium morphologies. In areas where the flow from rivers is highly regulated, it is important to give decision makers the ability to determine optimal flow management schemes, in order to allow tidal inlets to function as naturally as possible, and minimise the risk of inlet closure. The River Murray Mouth in South Australia is one such problem area. Drought and the retention of water for irrigation and urban water consumption have limited the amount of water entering the estuary. As a result, sediment from the coastal environment is being deposited in the mouth of the estuary, reducing the effect of further coastal interactions. Currently, situations such as this are modelled using traditional process-based methods, where wave, current, sediment transport and sediment balance modules are linked together in a time-stepping process. The modules are reapplied and assessed until a stable morphology is formed. In this paper, new options for modelling equilibrium morphologies of tidal inlets are detailed, which alleviate some of the shortfalls of traditional process-based models, such as the amplification of small errors and reliance on initial conditions. The modelling problem is approached in this paper from a different angle and involves the use of entropy based objective functions, which are optimised in order to find equilibrium morphologies. In this way, characteristics of a system at equilibrium can be recognised and a stable system predicted without having to step through time. This paper also details the use of self-organisation based modelling methods, another non-traditional model application, where local laws and feedback result in the formation of a global stable equilibrium morphology. These methods represent a different approach to traditional models, without some of the characteristics that may add to their limitations.

Compression of time‐generated matrices in two‐dimensional time‐domain elastodynamic BEM analysis

AbstractThis paper describes a new scheme to improve the efficiency of time‐domain BEM algorithms. The discussion is focused on the two‐dimensional elastodynamic formulation, however, the ideas presented apply equally to any step‐by‐step convolution based algorithm whose kernels decay with time increase. The algorithm presented interpolates the time‐domain matrices generated along the time‐stepping process, for time‐steps sufficiently far from the current time. Two interpolation procedures are considered here (a large number of alternative approaches is possible): Chebyshev–Lagrange polynomials and linear. A criterion to indicate the discrete time at which interpolation should start is proposed. Two numerical examples and conclusions are presented at the end of the paper. Copyright © 2004 John Wiley &amp; Sons, Ltd.

A HYBRID PSEUDO-FORCE/LAPLACE TRANSFORM METHOD FOR NON-LINEAR TRANSIENT RESPONSE OF A SUSPENDED CABLE

A hybrid numerical scheme involving the combination of the Laplace transform technique and the pseudo-force method is proposed to analyze the non-linear transient response of a suspended cable subjected to arbitrary dynamic loading. A theoretical model of the cable with multi-degree-of-freedom is first obtained through discretization of the partial differential equations by finite difference approximation. The non-linear governing equations take into account the effects of quadratic and cubic geometric non-linearities. The proposed method deals with the non-linear effects as pseudo-forces and then establishes an iterative solution scheme in the alternating Laplace/time domain by means of fast numerical Laplace transform. This method eschews a time-stepping process and therefore is computationally efficient. It also readily deals with the viscoelastic damping with frequency-dependent model parameters and the hysteresis damping in terms of complex stiffness models. Numerical examples are presented to evaluate the dynamic responses of suspended cables under a concentrated sinusoidal force and a distributed random excitation, and to identify the non-linear response properties by comparison with the linear vibration. The validity and accuracy of the proposed method is also verified by comparing the results with those obtained by using the direct time integration.

An evaluation of the gradient-weighted moving-finite-element method in one space dimension

Moving-grid methods are becoming increasingly popular for solving several kinds of parabolic and hyperbolic partial differential equations involving fine-scale structures such as steep moving fronts and emerging steep layers. An interesting example of such a method is provided by the moving-finite-element (MFE) method. A difficulty with MFE, as with many other existing moving-grid methods, is the threat of grid distortion, which can only be avoided by the use of penalty terms. The involved parameter tuning is known to be very important, not only to provide for a safe automatic grid-point selection, but also for efficiency in the time-stepping process. When compared with MFE, the gradient-weighted MFE (GWMFE) method has some promising properties to reduce the need of tuning. To investigate to what extent GWM FE can be called robust, reliable, and effective for the automatic solution of time-dependent PDEs in one space dimension, we have tested this method extensively on a set of five relevant example problems with various solution characteristics. All tests have been carried out using the BDF time integrator SPGEAR of the existing method-of-lines software package SPRINT.

An Efficient Interpolation Procedure for High-Order Three-Dimensional Semi-Lagrangian Models

An efficient method is proposed for performing the grid interpolations required at each advective time step of a multilevel, limited-area semi-Lagrangian model. The distinctive feature of the method is that it is composed of a cascade of one-dimensional interpolations of entire fields of data through a sequence of intermediate grids. These intermediate grids are formed as hybrid combinations of the standard model grid coordinates, which, together with the Lagrangian coordinates, are delineated by the origins of the trajectories that characterize the semi-Lagrangian method. When applied at Nth-order accuracy, the cascade method requires only O(N) operations compared with O(N3) for the conventional three-dimensional interpolation methods, making the adoption of high-order schemes attractive. The technique was tested on a large number (100) of 48-h forecasts and was found to be as accurate as the conventional interpolation procedures based on point-by-point Cartesian products of one-dimensional inter-polators. However, the cascade interpolation technique was 2.9, 6.1, and 10.2 times as fast as the conventional interpolation scheme for fourth-, sixth-, and eighth-order, respectively. We observe that the cascade method is equally applicable to the problem of interpolation from the grid of termini of forward trajectories to the standard model grid, for which there is no obvious counterpart in the Cartesian product method. Our technique therefore opens the way to a whole new class of high-order accurate semi-Lagrangian methods that incorporate the use of forward trajectories as part of the time-stepping process.