Uniform Time Step Research Articles

A time-dependent surrogate model for storm surge prediction based on an artificial neural network using high-fidelity synthetic hurricane modeling

Expedient prediction of storm surge is required for emergency managers to make critical decisions for evacuation, structure closure, and other emergency responses. However, time-dependent storm surge models do not exist for fast and accurate prediction in very short periods on the order of seconds to minutes. In this paper, a time-dependent surrogate model of storm surge is developed based on an artificial neural network with synthetic simulations of hurricanes. The neural network between six input hurricane parameters and one target parameter, storm surge, is trained by a feedforward backpropagation algorithm at each of 92 uniform time steps spanning 45.5 h for each storm. The basis data consist of 446 tropical storms developed from a joint probability model that was based on historical tropical storm activity in the Gulf of Mexico. Each of the 446 storms was modeled at high fidelity using a coupled storm surge and nearshore wave model. Storm surge is predicted by the 92 trained networks for approaching hurricane climatological and track parameters in a few seconds. Furthermore, the developed surrogate model is validated with measured data and high-fidelity simulations of two historical hurricanes at four points in southern Louisiana. In general, the neural networks at or near the boundary between land and ocean are well trained and model predictions are of similar accuracy to the basis modeling suites. Networks based on modeling results from complex inland locations are relatively poorly trained.

Computationally efficient multi-time-step method for partitioned time integration of highly nonlinear structural dynamics

An efficient and accurate method for solving large-scale problems in non-linear structural dynamics is presented. The method uses dual-Schur domain decomposition to divide a large finite element mesh into a number of smaller subdomains, which are solved independently using a suitable mesh-size and time-step to capture the local spatial and temporal scales of the problem. Continuity of the solution between subdomains is enforced by Lagrange multipliers. It is shown that the proposed method is stable, accurate and computationally more efficient than using a uniform time-step for the entire mesh. Numerical examples are presented to illustrate and corroborate these properties.

Numerical analysis of the hydroelastic behavior of a vertical plate due to solitary waves

Interactions of a vertical elastic plate with fully nonlinear water waves were simulated. Utilizing the mixed Eulerian Lagrangian method for the free-surface flow and the finite element method for the deflection of an elastic plate, a fully coupled scheme for accurately determining fluid–plate motions was developed. Using this scheme, some modifications to the solvers for both fluid and plate were made. A hybrid wave-absorbing beach was installed to prevent wave reflection from the end of the wave tank. A fourth-order Runge–Kutta time-marching scheme with a uniform time step was applied to achieve numerical stability. The method was validated by simulating the wave generated by the initial deformation of a vertical plate and comparing the result with the corresponding analytical solution. For further validation, the hydroelastic behavior of a vertical plate induced by a pulse-type wave (where the initial pulse-type elevation of the free surface is specified) was computed, and the result was compared with another numerical result from a mode-expansion method. The interaction of a surface-piercing plate with nonzero initial free surface was then simulated, and the result was compared with the corresponding linear analytical solution. Finally, the hydroelastic response of a surface-piercing vertical plate due to a solitary wave (generated by actuating the vertical plate at the right end of the tank only at the beginning) was computed and investigated systematically.

VARIABLE TIME-STEPPING HYBRID FINITE DIFFERENCE METHODS FOR PRICING BINARY OPTIONS

Two types of new methods with variable time steps are proposed in order to valuate binary options efficiently. Type I changes adaptively the size of the time step at each time based on the magnitude of the local error, while Type II combines two uniform meshes. The new methods are hybrid finite difference methods, namely starting the computation with a fully implicit finite difference method for a few time steps for accuracy then performing a <TEX>${\theta}$</TEX>-method during the rest of computation for efficiency. Numerical experiments for standard European vanilla, binary and American options show that both Type I and II variable time step methods are much more efficient than the fully implicit method or hybrid methods with uniform time steps.

Nonlinear analysis on wave-plate interaction due to disturbed vertical elastic plate

Hydroelastic behavior of wave-plate interaction due to an initial deflection of a vertical elastic plate is simulated by using a Mixed Eulerian Lagrangian method for the fluid and a Finite Element Method for the plate. An implicit boundary-condition method is developed to solve the coupled motions of fluid and plate. A hybrid wave absorbing beach is installed to prevent the wave reflection from a vertical wall at the end of the wave tank. The 4th order Runge-Kutta scheme with uniform time step is used for time marching. First, numerical results of the elastic vibration of a vertical plate induced by deformation of itself at initial time are validated by comparing with corresponding analytical solution. Then, further comparison between the results from mode-expansion method and FEM is made. Finally, the influences of the edge condition at the top of plate and the plate stiffness on the wave-plate interaction are investigated systematically.

A modified truncation error-based adaptive time-step control strategy for fully- and semi-implicit simulation of isothermal solidification processes

When analyzing the transient characteristics of solidification processes, choosing appropriately-sized time steps is difficult. Accordingly, the current study develops a modified local time truncation error (LTE)-based strategy designed to adaptively adjust the size of the time step during the simulated solidification procedure in such a way that the time steps can be adapted in accordance with the local variations in latent heat released during phase change or the effects of pure conduction in a single solid or liquid phase. In the approach presented in this work, the LTE-based time-step evaluation procedure is applied not only after a convergent temperature field is obtained at each time step, but also during the nonlinear iterations performed at each time step whenever a convergence problem is encountered. The computational accuracy and efficiency of the proposed method are demonstrated via the simulation of the one-dimensional and two-dimensional solidification problems and compared with those of other adaptive time step and the uniform time step methods. Furthermore, the performance of these approaches has also been demonstrated using fully-implicit and semi-implicit schemes.

Adaptive lattice methods for multi-asset models

Adaptive lattice methods are developed to compute the price of multivariate contingent claims. A simple coordinate representation is used to extend one dimensional lattice methods to multivariate asset models. Two algorithms are proposed, one performing several levels of refinement for a time interval [ T − Δ t , T ] and the other performing one level of refinement for λ % of a given time domain [ 0 , T ] , where T is the time to maturity, Δ t is the time step size and λ > 0 is a constant. Numerical experiments are carried out for the European and American barrier-type options with one, two, or three underlying assets. In our numerical experiments, both adaptive algorithms improve efficiency over lattice methods with a uniform time step for the same level of accuracy.

Quantum and electromagnetic propagation with the conjugate symmetric Lanczos method

The conjugate symmetric Lanczos (CSL) method is introduced for the solution of the time-dependent Schrodinger equation. This remarkably simple and efficient time-domain algorithm is a low-order polynomial expansion of the quantum propagator for time-independent Hamiltonians and derives from the time-reversal symmetry of the Schrodinger equation. The CSL algorithm gives forward solutions by simply complex conjugating backward polynomial expansion coefficients. Interestingly, the expansion coefficients are the same for each uniform time step, a fact that is only spoiled by basis incompleteness and finite precision. This is true for the Krylov basis and, with further investigation, is also found to be true for the Lanczos basis, important for efficient orthogonal projection-based algorithms. The CSL method errors roughly track those of the short iterative Lanczos method while requiring fewer matrix-vector products than the Chebyshev method. With the CSL method, only a few vectors need to be stored at a time, there is no need to estimate the Hamiltonian spectral range, and only matrix-vector and vector-vector products are required. Applications using localized wavelet bases are made to harmonic oscillator and anharmonic Morse oscillator systems as well as electrodynamic pulse propagation using the Hamiltonian form of Maxwell's equations. For gold with a Drude dielectric function, the latter is non-Hermitian, requiring consideration of corrections to the CSL algorithm.

An Adaptive Time-Step Control Strategy for the Solidification Processes Based on Modified Local Time Truncation Error

Choosing appropriate time steps to model the transient and discontinuous characteristics of solidification processes is difficult. The current study develops a modified local time truncation error (LTE)-based strategy designed to adaptively adjust the size of the time step during the simulated solidification procedure in such a way that the time steps can be adapted in accordance with the local variations in latent heat released during phase change or the effects of pure conduction in a single solid or liquid phase. The computational accuracy, efficiency and convergence of the proposed method are demonstrated via the simulation of the one-dimensional and two-dimensional solidification problems and compared with those of other uniform time step and adaptive time step methods. Consequently, the effects of latent heat release are more accurately modeled, the precision and efficiency of the computational solutions is correspondingly improved, and the computational errors are minimized. Furthermore, in solving the 2-D problem, it is shown that the line Gauss-Seidel iteration method and the proposed nonlinear iteration method can be combined to construct a highly efficient and accurate solver.

Structure and Evolution of Nearby Stars with Planets. II. Physical Properties of ∼1000 Cool Stars from the SPOCS Catalog

We derive detailed theoretical models for 1074 nearby stars from the SPOCS (Spectroscopic Properties of Cool Stars) Catalog. The California and Carnegie Planet Search has obtained high-quality (R 70,000-90,000, S/N 300-500) echelle spectra of over 1000 nearby stars taken with the Hamilton spectrograph at Lick Observatory, the HIRES spectrograph at Keck, and UCLES at the Anglo Australian Observatory. A uniform analysis of the high-resolution spectra has yielded precise stellar parameters (Teff, log g, v sin i, [M/H], and individual elemental abundances for Fe, Ni, Si, Na, and Ti), enabling systematic error analyses and accurate theoretical stellar modeling. We have created a large database of theoretical stellar evolution tracks using the Yale Stellar Evolution Code (YREC) to match the observed parameters of the SPOCS stars. Our very dense grids of evolutionary tracks eliminate the need for interpolation between stellar evolutionary tracks and allow precise determinations of physical stellar parameters (mass, age, radius, size and mass of the convective zone, surface gravity, etc.). Combining our stellar models with the observed stellar atmospheric parameters and uncertainties, we compute the likelihood for each set of stellar model parameters separated by uniform time steps along the stellar evolutionary tracks. The computed likelihoods are used for a Bayesian analysis to derive posterior probability distribution functions for the physical stellar parameters of interest. We provide a catalog of physical parameters for 1074 stars that are based on a uniform set of high-quality spectral observations, a uniform spectral reduction procedure, and a uniform set of stellar evolutionary models. We explore this catalog for various possible correlations between stellar and planetary properties, which may help constrain the formation and dynamical histories of other planetary systems.

Behavioral simulation of fractional-N frequency synthesizers and other PLL circuits

Two techniques are presented that allow fast and accurate simulation of fractional-N synthesizers. A uniform time step allows implementation of these techniques in various simulation frameworks, such as Verilog, Matlab, and C or C++ programs. The techniques are also applicable to the simulation of other PLL systems, such as clock and data recovery circuits.

Three-dimensional simulation of high compressible flows using a multi-time-step integration technique with subcycles

An algorithm to simulate three-dimensional high compressible flows using the finite element method and a multi-time-step integration technique with subcycles is presented in this work. An explicit two-step Taylor–Galerkin scheme is adopted to integrate in time the continuum equations. When explicit schemes are used, the time-steps must satisfy the CFL stability conditions. If the smallest critical time-step is adopted uniformly for the whole domain, the integration scheme may consume a large amount of CPU time. Multi-time-step integration techniques are very suitable in these cases because elements and nodes are separated into several groups and a different time-step is assigned to each group. In this way, each group of elements is integrated with a time interval which is much closer to the critical time-steps of the elements in the group. This results in great computational savings, mainly when element sizes and properties are very different, leading to significant differences in the local critical time-step values. Multi-time-steps integration techniques are also very useful in transient problems, taking into account that at the end of each subcycle, values of the unknowns at the same time level are obtained. The multi-time-step algorithm is applied to analyze the supersonic flow (Mach=8.5) past a sphere immersed in a non-viscous flow, and the results and computational performance are compared with those obtained when a uniform time-step is used over the whole domain.

Adaptive finite elements for a linear parabolic problem

A posteriori error estimates for the heat equation in two space dimensions are presented. A classical discretization is used, Euler backward in time, and continuous, piecewise linear triangular finite elements in space. The error is bounded above and below by an explicit error estimator based on the residual. Numerical results are presented for uniform triangulations and constant time steps. The quality of our error estimator is discussed. An adaptive algorithm is then proposed. Successive Delaunay triangulations are generated, so that the estimated relative error is close to a preset tolerance. Again, numerical results demonstrate the efficiency of our approach.

AP1--P1Finite Element Method for a Phase Relaxation Model I: Quasi-Uniform Mesh

We study a simple model of phase relaxation which consists of a parabolic PDE for temperature $\theta$ and an ODE with a small parameter $\ep$ and double obstacles for phase variable $\chi$. The model replaces sharp interfaces by diffuse ones and gives rise to superheating effects. A semi-explicit time discretization with uniform time step $\tau$ is combined with continuous piecewise linear finite elements for both $\theta$ and $\chi$, over a fixed quasi-uniform mesh of size h. At each time step, an inexpensive nodewise algebraic correction is performed to update $\chi$, followed by the solution of a linear positive definite symmetric system for $\theta$ by a preconditioned conjugate gradient method. A priori estimates for both $\theta$ and $\chi$ are derived in L2-based Sobolev spaces provided the stability constraint $\tau\leq\ep$ is enforced. Asymptotic behavior of the fully discrete model is examined as $\ep,\tau,h\da0$ independently, which leads to a rate of convergence of order $\O((\tau+h)\ep^{-1/2})$, provided a natural compatibility condition on the initial data is satisfied. Numerical experiments illustrate the performance of the proposed method for the natural choice $h\approx\tau\leq\ep$.

Convergence Past Singularities for a Fully Discrete Approximation of Curvature-Driven Interfaces

Consider a closed surface in $\Rn$ of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term. This geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter $\eps>0$. Conforming piecewise linear finite elements over a quasi-uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time-step $\tau$. It is shown that the zero level set of the fully discrete solution converges past singularities to the true interface, provided $\tau,h^2\approx o(\eps^3)$ and no fattening occurs. If the more stringent relations $\tau,h^2\approx\O(\eps^4)$ are enforced, then a linear rate of convergence $\O(\eps)$ for interfaces is derived in the vicinity of regular points, namely those for which the underlying viscosity solution is nondegenerate. Singularities and their smearing effect are also studied. The analysis is based on constructing discrete barriers via a parabolic projection, Lipschitz dependence of viscosity solutions with respect to perturbations of data, and discrete nondegeneracy. These issues are proven, along with quasi optimality in two dimensions of the parabolic projection in $L^\infty$ with respect to both order and regularity requirements for functions in $W^{2,1}_p$.

Combined effect of explicit time-stepping and quadrature for curvature driven flows

The flow of a closed surface of codimension 1 in ${\Bbb R}^n$ driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter $\varepsilon>0$ . Conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh of size $h$ are further used for space discretization, and combined with forward differences for time discretization with uniform time-step $\tau$ . The resulting explicit schemes are the basis for an efficient algorithm, the so-called dynamic mesh algorithm, and exhibit finite speed of propagation and discrete ondegeneracy. No iteration is required, not even to handle the obstacle constraints. The zero level set of the fully discrete solution is shown to converge past singularities to the true interface, provided $\tau,h^2\approx o(\varepsilon^4)$ and no fattening occurs. If the more stringent relations $\tau,h^2\approx O(\varepsilon^6)$ are enforced, then an interface rate of convergence $O(\varepsilon)$ is derived in the vicinity of regular points, along with a companion $O(\varepsilon^{1/2})$ for type I singularities. For smooth flows, an interface rate of convergence of $O(\varepsilonq)$ is proven, provided $\tau,h^2 \approx O(\varepsilon^5)$ and exact integration is used for the potential term. The analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature, which bears some intrinsic interest, Lipschitz properties of viscosity solutions of the level set approach, and discrete nondegeneracy. These basic ingredients are also discussed.

A Fully Discrete Adaptive Nonlinear Chernoff Formula

An adaptive piecewise linear finite element approximation to a linear method, the so-called nonlinear Chernoif formula, for the simplest two-phase Stefan problem in two dimensions are discussed. The full discretization amounts to solving linear positive definite symmetric systems, followed by an explicit nodewise algebraic correction. Consecutive meshes are highly graded and noncompatible. The pointwise information needed to generate a new mesh is extracted from both the discrete enthalpy $U^n $ and temperature $\Theta ^n $. For well-behaved discrete transition layers $\mathcal{T}^n = \{ 0 < U^n - \Theta ^n < 1\} $, triangles are expected to be $\mathcal{O}(\tau )$ within a thin strip of width $\mathcal{O}(\tau ^{{1 / 2}} )$, the so-called refined region, which in turn must contain the transition layer. The local meshsize increases up to $\mathcal {O}(\tau ^{1/2} )$ elsewhere; here $\tau $ stands for the uniform time-step. The expected number of spatial degrees of freedom becomes $\mathcal {O}(\tau ^{ - 3/2} )$, which compares quite favorably with that required without adaptivity, namely $\mathcal {O}(\tau ^{ - 2} )$. Stability is examined in a number of Sobolev norms and then is used to derive a quasi-optimal rate of convergence of order essentially $\mathcal {O}(\tau ^{1/2} )$ in the natural energy spaces. Numerical experiments illustrate the performance of the proposed method.

An adaptive finite element method for two-phase Stefan problems in two space dimensions. I. Stability and error estimates

A simple and efficient adaptive local mesh refinement algorithm is devised and analyzed for two-phase Stefan problems in 2D. A typical triangulation is coarse away from the discrete interface, where discretization parameters satisfy a parabolic relation, whereas it is locally refined in the vicinity of the discrete interface so that the relation becomes hyperbolic. Several numerical tests are performed on the computed temperature to extract information about its first and second derivatives as well as to predict discrete free boundary locations. Mesh selection is based upon equidistributing pointwise interpolation errors between consecutive meshes and imposing that discrete interfaces belong to the so-called refined region. Consecutive meshes are not compatible in that they are not produced by enrichment or coarsening procedures but rather regenerated. A general theory for interpolation between noncompatible meshes is set up in ${L^p}$-based norms. The resulting scheme is stable in various Sobolev norms and necessitates fewer spatial degrees of freedom than previous practical methods on quasi-uniform meshes, namely $O({\tau ^{ - 3/2}})$ as opposed to $O({\tau ^{ - 2}})$, to achieve the same global asymptotic accuracy; here $\tau > 0$ is the (uniform) time step. A rate of convergence of essentially $O({\tau ^{1/2}})$ is derived in the natural energy spaces provided the total number of mesh changes is restricted to $O({\tau ^{ - 1/2}})$, which in turn is compatible with the mesh selection procedure. An auxiliary quasi-optimal pointwise error estimate for the Laplace operator is proved as well. Numerical results illustrate the scheme’s efficiency in approximating both solutions and interfaces.