Temporal Grid Research Articles

A second-order box solver for nonlinear delayed convection-diffusion equations with Neumann boundary conditions

ABSTRACTIn this paper, by applying order reduction approach, a second-order accurate box scheme is established to solve a nonlinear delayed convection-diffusion equations with Neumann boundary conditions. By the discrete energy method, it is shown that the difference scheme is uniquely solvable, and has a convergence rate of with respect to - norm in constrained and non-constrained temporal grids. Besides, for constrained temporal step, a Richardson extrapolation method (REM) used along with the box scheme, which makes final solution third-order accurate in both time and space, is developed in detail. Finally, numerical results confirm the accuracy and efficiency of our solvers.

An RBF–FD method for pricing American options under jump–diffusion models

American option prices under jump–diffusion models are determined as solutions to partial integro-differential equations (PIDE). In this paper a new combination of a time and spatial discretization applied to a linear complementary formulation (LCP) of the free boundary PIDE is proposed. First a coordinate stretching transformation is performed to the asset price so that the computation of the prices can be focused on regions of real interest instead of on the whole solution domain. An implicit–explicit time discretization applied to the reformulated LCP on a uniform temporal grid is followed by a spatial discretization to get a fully discrete system. The radial basis function (RBF) finite difference method is a local method resulting in a sparse linear system in contrast to global RBF-methods which lead to ill-conditioned dense matrix systems. For the corresponding European option we prove consistency, stability and second-order convergence in a discrete L2-norm. We derive mild conditions for the model parameters under which these results hold. Numerical experiments are performed with European and American options, and a comparison with numerical results available in the literature illustrates the accuracy and efficiency of the proposed algorithm.

Nonlinear simulations of thermo-resistive tearing mode formalism of the density limit

This work is an extension of the previous semi-analytical work (Teng et al 2016 Nucl. Fusion 56 106001), explaining the density limit with a thermo-resistive tearing mode model. The thermo-resistive island growth is calculated with a 3D MHD code M3D-C1 (Jardin et al 2012 Comput. Sci. Discovery 5 014002). It is shown with nonlinear 3D MHD simulations that impurity radiation stimulates large magnetic islands at the density limit. The impact of thermal perturbations inside and outside of the island is explored. Inside the island, net cooling enhances the island growth and net heating suppresses its growth. Outside the island, thermal perturbations have a much smaller impact on the island growth. When the plasma density is increased towards the density limit, the island changes from being heated to being cooled and grows to a much larger island width. The convergence test of the simulations over the temporal and spatial grid is performed. The numerical model and the semi-analytical model are compared by calculating the terms, which are defined in the modified Rutherford equation, and good agreement is observed.

Wave Climate at Shallow Waters along the Abu Dhabi Coast

High-resolution, reliable global atmospheric and oceanic numerical models can represent a key factor in designing a coastal intervention. At the present, two main centers have the capabilities to produce them: the National Oceanic and Atmospheric Administration (NOAA) in the U.S.A. and the European Centre for Medium-Range Weather Forecasts (ECMWF). The NOAA and ECMWF wave models are developed, in particular, for different water regions: deep, intermediate, and shallow water regions using different types of spatial and temporal grids. Recently, in the Arabian Gulf (also named Persian Gulf), the Abu Dhabi Municipality (ADM) installed an ADCP (Acoustic Doppler Current Profiler) to observe the atmospheric and oceanographic conditions (water level, significant wave height, peak wave period, water temperature, and wind speed and direction) at 6 m water depth, in the vicinity of the shoreline of the Saadiyat beach. Courtesy of Abu Dhabi Municipality, this observations dataset is available; the recorded data span the period from June 2015 to January 2018 (included), with a time resolution of 10 min and 30 min for the atmospheric and oceanographic variables, respectively. At the ADCP deployment location (ADMins), the wave climate has been determined using wave propagation of the NOAA offshore wave dataset by means of the Simulating WAves Nearshore (SWAN) numerical model, the NOAA and ECMWF wave datasets at the closest grid point in shallow water conditions, and the SPM ’84 hindcasting method with the NOAA wind dataset used as input. It is shown that the best agreement with the observed wave climate is obtained using the SPM ’84 hindcasting method for the shallow water conditions.

The high-order multistep ADI solver for two-dimensional nonlinear delayed reaction–diffusion equations with variable coefficients

In this paper, a high-order compact multistep alternating direction implicit (ADI) difference scheme is constructed and analyzed for two-dimensional (2D) nonlinear delayed reaction–diffusion equations (NDREs) with variable coefficients. By the discrete energy method, it is presented that it achieves the convergence rate of O(τ2+hx4+hy4) with respect to H1- and L2-norms in non-constrained temporal grids, and is unconditionally stable. Finally, numerical results demonstrate the effectiveness of our ADI solver and exhibit the correctness of theoretical results.

A compact adaptive approach for degenerate singular reaction‐diffusion equations

This article concerns a compact adaptive method for the numerical solution of nonlinear degenerate singular reaction‐diffusion equations. The partial differential equation problems exhibit strong quenching blow‐up type singularities, and are critical in numerous applications ranging from optimized internal combustion designs to oil pipeline decay predictions. Adaptive schemes of fourth order in space and second order in time are acquired and discussed. Nonuniform spatial and temporal grids are utilized through suitable adaptations. Rigorous analysis is given for the numerical stability, and computational experiments are performed to illustrate our conclusions.

Convergence of the multigrid reduction in time algorithm for the linear elasticity equations

SummaryThis paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.

A block-centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system

A block-centered finite difference method is proposed for solving an unsteady asymptotic coupled model, in which the flow is governed by Darcy’s law both in the one-dimensional fracture and two-dimensional porous media. The second-order error estimates in discrete norms are derived on nonuniform rectangular grids for both pressure and velocity. The numerical scheme can be extended to nonmatching spatial and temporal grids without loss of accuracy. Numerical experiments are performed to verify the efficiency and accuracy of the proposed method. It is shown that the pressure and velocity are discontinuous across the fracture-interface and the fracture indeed acts as the fast pathway or geological barrier in the aquifer system.

Simulation of the flow around a circular cylinder at Re=3900 with Partially-Averaged Navier-Stokes equations

This study employs Partially-Averaged Navier-Stokes (PANS) equations to simulate the flow around a smooth circular cylinder at Reynolds number 3900. It intends to evaluate the importance of discretization and modelling errors on the accuracy of this mathematical model. Furthermore, the study addresses the effect of the physical resolution, or fraction of turbulence kinetic energy being modelled fk, on the predictions accuracy. To this end, Validation exercises are carried out using five different values of fk which range from typical values for well-resolved Scale-Resolving Simulations (fk ≤ 0.25) to Reynolds-Averaged Navier-Stokes equations (fk=1.00). Naturally, these exercises require the evaluation of numerical errors, i.e. Verification studies. Consequently, and taking advantage of the ability of PANS to enable the distinction between discretization and modelling errors, spatial and temporal grid refinement studies are carried out to assess the magnitude of the discretization error, as well as its dependence on fk. The outcome confirms the ability of PANS, in combination with fk < 0.50, to substantially decrease the modelling error when compared to fk=1.00. However, the reduction of fk tends to increase the model dependence on the spatial and temporal resolution. It is demonstrated that similarly to the effect of the spatial and temporal grid resolution on the magnitude of the numerical error, the modelling error diminishes with the physical resolution (fk → 0). The convergence of the predictions with fk is also illustrated.

Smooth Quintic spline approximation for nonlinear Schrödinger equations with variable coefficients in one and two dimensions

The present paper uses a relatively new approach and methodology to solve one and two dimensional nonlinear Schrödinger equations numerically. We use the horizontal method of lines and θ-method, θ ∈ [1/2, 1] for time discretization that reduces the problem into an amenable system of ordinary differential equations. The resulting system of ODEs in space subsequently have been solved by quintic polynomial spline scheme. Convergence of the scheme in maximum norm is established rigorously. The convergence orders are O(k+hx4+hy4) and O(k2+hx4+hy4), where k is the temporal grid size and hx and hy are spatial grid sizes, respectively. Matrix stability analysis shows that the method is conditionally stable. The efficacy of proposed approach has been confirmed with four numerical experiments, where comparison is made with some earlier works. It is clear that the results obtained are acceptable and are in good agreement with earlier studies. The present scheme is very simple, effective and convenient for obtaining numerical solution of Schrödinger equation.

Numerical Solution of the Time Dependent 3D Schrödinger Equation Describing Tunneling of Atoms from Anharmonic Traps

We present an efficient numerical method for the integration of the 3D Schrödinger equation. A tunneling problem of two interacting bosonic atoms confined in a 1D anharmonic trap has been successfully solved by means of this method. We demonstrate fast convergence of the final results with respect to spatial and temporal grid steps. The computational scheme is based on the operator-splitting technique with the implicit Crank-Nicolson algorithm on spatial sixth-order finite-differences. The computational time is proportional to the number of spatial grid points.

Relationships between early literacy and nonlinguistic rhythmic processes in kindergarteners

A growing number of studies report links between nonlinguistic rhythmic abilities and certain linguistic abilities, particularly phonological skills. The current study investigated the relationship between nonlinguistic rhythmic processing, phonological abilities, and early literacy abilities in kindergarteners. A distinctive aspect of the current work was the exploration of whether processing of different types of rhythmic patterns is differentially related to kindergarteners’ phonological and reading-related abilities. Specifically, we examined the processing of metrical versus nonmetrical rhythmic patterns, that is, patterns capable of being subdivided into equal temporal intervals or not (Povel & Essens, 1985). This is an important comparison because most music involves metrical sequences, in which rhythm often has an underlying temporal grid of isochronous units. In contrast, nonmetrical sequences are arguably more typical to speech rhythm, which is temporally structured but does not involve an underlying grid of equal temporal units. A rhythm discrimination app with metrical and nonmetrical patterns was administered to 74 kindergarteners in conjunction with cognitive and preliteracy measures. Findings support a relationship among rhythm perception, phonological awareness, and letter-sound knowledge (an essential precursor of reading). A mediation analysis revealed that the association between rhythm perception and letter-sound knowledge is mediated through phonological awareness. Furthermore, metrical perception accounted for unique variance in letter-sound knowledge above all other language and cognitive measures. These results point to a unique role for temporal regularity processing in the association between musical rhythm and literacy in young children.

Critical Points and Traveling Wave in Locomotion: Experimental Evidence and Some Theoretical Considerations.

The central pattern generator (CPG) architecture for rhythm generation remains partly elusive. We compare cat and frog locomotion results, where the component unrelated to pattern formation appears as a temporal grid, and traveling wave respectively. Frog spinal cord microstimulation with N-methyl-D-Aspartate (NMDA), a CPG activator, produced a limited set of force directions, sometimes tonic, but more often alternating between directions similar to the tonic forces. The tonic forces were topographically organized, and sites evoking rhythms with different force subsets were located close to the constituent tonic force regions. Thus CPGs consist of topographically organized modules. Modularity was also identified as a limited set of muscle synergies whose combinations reconstructed the EMGs. The cat CPG was investigated using proprioceptive inputs during fictive locomotion. Critical points identified both as abrupt transitions in the effect of phasic perturbations, and burst shape transitions, had biomechanical correlates in intact locomotion. During tonic proprioceptive perturbations, discrete shifts between these critical points explained the burst durations changes, and amplitude changes occurred at one of these points. Besides confirming CPG modularity, these results suggest a fixed temporal grid of anchoring points, to shift modules onsets and offsets. Frog locomotion, reconstructed with the NMDA synergies, showed a partially overlapping synergy activation sequence. Using the early synergy output evoked by NMDA at different spinal sites, revealed a rostrocaudal topographic organization, where each synergy is preferentially evoked from a few, albeit overlapping, cord regions. Comparing the locomotor synergy sequence with this topography suggests that a rostrocaudal traveling wave would activate the synergies in the proper sequence for locomotion. This output was reproduced in a two-layer model using this topography and a traveling wave. Together our results suggest two CPG components: modules, i.e., synergies; and temporal patterning, seen as a temporal grid in the cat, and a traveling wave in the frog. Animal and limb navigation have similarities. Research relating grid cells to the theta rhythm and on segmentation during navigation may relate to our temporal grid and traveling wave results. Winfree’s mathematical work, combining critical phases and a traveling wave, also appears important. We conclude suggesting tracing, and imaging experiments to investigate our CPG model.

Biometric Differences between Unilateral Chronic Primary Angle Closure Glaucoma and Fellow Non-Glaucomatous Eyes

ABSTRACTPurpose: To compare biometric differences between eyes with unilateral chronic primary angle-closure glaucoma (PACG) and fellow non-glaucomatous eyes in the same patient. Methods: Clinical data and imaging records of 17 patients with unilateral PACG were retrospectively reviewed. The fellow eyes with primary angle-closure (PAC) or primary angle-closure suspected (PACS) were grouped in Group 1. The PACG eyes were set as Group 2. The biometric parameters of both eyes were measured by IOL-Master, anterior segment optical coherence tomography (AS-OCT), and swept source OCT. Results: The iris area in Group 1 was significantly thicker than that in Group 2 (1.590 ± 0.267 versus 1.365 ± 0.352, P = 0.016). Choroidal thickness in the macular region was thicker in Group 1 than in Group 2, with statistical significance at inner temporal grid (282.7 ± 121.1 versus 191.6 ± 90.3, P = 0.036), central field (297.4 ± 130.6 versus 200.1 ± 96.3, P = 0.029), inner nasal grid (283.1 ± 128.3 versus 194.8 ± 91.8, P = 0.040), and average value (265.3 ± 105.6 versus 191.1 ± 87.0, P = 0.049). Group 1 also had thicker peripapillary choroidal thickness at 1 o’clock (141.4 ± 68.4 versus 101.8 ± 39.0, P = 0.025) and 12 o’clock (141.5 ± 82.3 versus 104.5 ± 44.6, P = 0.037) compared to Group 2. The multivariate logistic regression analyses showed that only intraocular pressure was independently associated with PACG and explained 58.3% variance of PACG. Conclusions: In unilateral PACG, there was significant inter-eye difference in anterior and posterior segment parameters. Longitudinal comparisons are required to further understand the differences in pathology of angle closures.

On the computation for blow-up solutions of the nonlinear wave equation

We consider in this paper the 1-dim nonlinear wave equation $$\frac{\partial ^2 u}{\partial t^2}(t,x)=\frac{\partial ^2 u}{\partial x^2}(t,x)+|u|^{1+\alpha }(t,x)\;(\alpha >0)$$ and its finite difference analogue. It is known that the solution of the current equation may become unbounded in finite time, a phenomenon which is known as blow-up. Moreover, since the nonlinear wave equation enjoys finite speed of propagation, even if the solution has become unbounded at certain points, the solution continues to exist, blows up at later times and forms the so-called blow-up curve. Up to the present, however, numerical approximation for blow-up problems gave only the convergence up to the blow-up time of $$\Vert u(t,\cdot )\Vert _\infty $$ . In fact, adaptive time mesh strategy was used to numerically reproduce the phenomenon of finite-time blow-up. Nevertheless, computation using such schemes can not be extended beyond the blow-up time of $$\Vert u(t,\cdot )\Vert _\infty $$ . This is a fatal problem when computing blow-up solutions for the nonlinear wave equation. As a consequence, we reconsider a finite difference scheme whose temporal grid size is given uniformly and propose an algorithm with rigorous convergence analysis to numerically reconstruct the blow-up curve for the nonlinear wave equation.

A non-standard finite difference scheme for a diffusive HBV infection model with capsids and time delay

In this paper, a non-standard finite difference (NSFD) scheme for a delayed diffusive hepatitis B virus (HBV) infection model with intracellular HBV DNA-containing capsids is proposed. Dynamic consistency of this NSFD scheme is achieved by showing that the scheme preserves the non-negativity and boundedness of the solutions and the global stability of the homogeneous steady states of the corresponding continuous model without any restriction on spatial and temporal grid sizes. We prove the global stability of the steady states by constructing suitable discrete Lyapunov functions.

A transformed path integral approach for solution of the Fokker–Planck equation

A novel path integral (PI) based method for solution of the Fokker–Planck equation is presented. The proposed method, termed the transformed path integral (TPI) method, utilizes a new formulation for the underlying short-time propagator to perform the evolution of the probability density function (PDF) in a transformed computational domain where a more accurate representation of the PDF can be ensured. The new formulation, based on a dynamic transformation of the original state space with the statistics of the PDF as parameters, preserves the non-negativity of the PDF and incorporates short-time properties of the underlying stochastic process. New update equations for the state PDF in a transformed space and the parameters of the transformation (including mean and covariance) that better accommodate nonlinearities in drift and non-Gaussian behavior in distributions are proposed (based on properties of the SDE). Owing to the choice of transformation considered, the proposed method maps a fixed grid in transformed space to a dynamically adaptive grid in the original state space. The TPI method, in contrast to conventional methods such as Monte Carlo simulations and fixed grid approaches, is able to better represent the distributions (especially the tail information) and better address challenges in processes with large diffusion, large drift and large concentration of PDF. Additionally, in the proposed TPI method, error bounds on the probability in the computational domain can be obtained using the Chebyshev's inequality. The benefits of the TPI method over conventional methods are illustrated through simulations of linear and nonlinear drift processes in one-dimensional and multidimensional state spaces. The effects of spatial and temporal grid resolutions as well as that of the diffusion coefficient on the error in the PDF are also characterized.

Validation of sea ice models using an uncertainty‐based distance metric for multiple model variables

AbstractWe implement a variance‐based distance metric (Dn) to objectively assess skill of sea ice models when multiple output variables or uncertainties in both model predictions and observations need to be considered. The metric compares observations and model data pairs on common spatial and temporal grids improving upon highly aggregated metrics (e.g., total sea ice extent or volume) by capturing the spatial character of model skill. The Dn metric is a gamma‐distributed statistic that is more general than the χ2 statistic commonly used to assess model fit, which requires the assumption that the model is unbiased and can only incorporate observational error in the analysis. The Dn statistic does not assume that the model is unbiased, and allows the incorporation of multiple observational data sets for the same variable and simultaneously for different variables, along with different types of variances that can characterize uncertainties in both observations and the model. This approach represents a step to establish a systematic framework for probabilistic validation of sea ice models. The methodology is also useful for model tuning by using the Dn metric as a cost function and incorporating model parametric uncertainty as part of a scheme to optimize model functionality. We apply this approach to evaluate different configurations of the standalone Los Alamos sea ice model (CICE) encompassing the parametric uncertainty in the model, and to find new sets of model configurations that produce better agreement than previous configurations between model and observational estimates of sea ice concentration and thickness.

A low-rank control variate for multilevel Monte Carlo simulation of high-dimensional uncertain systems

Multilevel Monte Carlo (MLMC) is a recently proposed variation of Monte Carlo (MC) simulation that achieves variance reduction by simulating the governing equations on a series of spatial (or temporal) grids with increasing resolution. Instead of directly employing the fine grid solutions, MLMC estimates the expectation of the quantity of interest from the coarsest grid solutions as well as differences between each two consecutive grid solutions. When the differences corresponding to finer grids become smaller, hence less variable, fewer MC realizations of finer grid solutions are needed to compute the difference expectations, thus leading to a reduction in the overall work. This paper presents an extension of MLMC, referred to as multilevel control variates (MLCV), where a low-rank approximation to the solution on each grid, obtained primarily based on coarser grid solutions, is used as a control variate for estimating the expectations involved in MLMC. Cost estimates as well as numerical examples are presented to demonstrate the advantage of this new MLCV approach over the standard MLMC when the solution of interest admits a low-rank approximation and the cost of simulating finer grids grows fast.

Reminiscence Therapy Improved Recall of Autobiographical Memory

Reminiscence Therapy Improved Recall of Autobiographical Memory