Non-uniform Time Grid Research Articles

Operator-difference schemes on non-uniform grids for second-order evolutionary equations

Abstract The approximate solution of the Cauchy problem for second-order evolution equations is performed, first of all, using three-level time approximations. Such approximations are easily constructed and relatively uncomplicated to investigate when using uniform time grids. When solving applied problems numerically, we should focus on approximations with variable time steps. When using multilevel schemes on non-uniform grids, we should maintain accuracy by choosing appropriate approximations and ensuring stability of the approximate solution. In this paper, we construct unconditionally stable schemes of the first- and second-order accuracy on a non-uniform time grid for the approximate solution of the Cauchy problem for a second-order evolutionary equation. The novelty of the paper consists in the fact that these stability estimates are obtained without any restrictions on the magnitude of the step change and on the number of step changes. We use a special transformation of the original second-order differential-operator equation to a system of first-order equations. For the system of first-order equations, we apply standard two-level time approximations. We obtained stability estimates for the initial data and the right-hand side in finite-dimensional Hilbert space. Eliminating auxiliary variables leads to three-level schemes for the initial second-order evolution equation. Numerical experiments were performed for the test problem for a one-dimensional in space bi-parabolic equation. The accuracy and stability properties of the constructed schemes are demonstrated on non-uniform grids with randomly varying grid steps.

Adaptive control arc length-based time grid refinement control parameterisation method for unmanned hypersonic vehicle reentry trajectory optimisation

An adaptive control arc length-based trajectory optimisation approach, which combines the control vector parameterisation (CVP) method with Gauss time grid distribution and adaptive grid refinement strategy, is proposed to further improve the downrange of unmanned hypersonic vehicles. Firstly, the Gauss time grid strategy is introduced to obtain a nonuniform Gauss time grid to calculate the original control profiles. By employing the curve integral formula to analyse the control arc length, an adaptive time grid refinement concept is proposed and the refinement process is derived in detail. The initial time grid nodes are then adaptively subdivided and combined. Finally, the control arc length-based nonuniform time grid refinement optimisation method is implemented. Numerical tests are carried out on common aero hypersonic vehicles and comparisons are made with other CVP methods. The test results reveal that the proposed approach can efficiently improve the downrange and decrease the path constraint oscillation.

Global superconvergence analysis of nonconforming finite element method for time fractional reaction-diffusion problem with anisotropic data

In this paper, a class of two-dimensional (2-D) time fractional reaction-diffusion equation is considered. The solution usually exhibits singularity at the initial moment and anisotropic behavior in the spatial direction. In response to these problems, we provide an effective numerical framework for analyzing the L2-norm error, H1-norm superclose property and H1-norm global superconvergence result. This framework combines the high-precision L2-1σ scheme on non-uniform time grids and the anisotropic nonconforming quasi-Wilson finite element method (FEM) in space. Some numerical experiments are presented to illustrate our theoretical findings.

A second-order scheme with nonuniform time grids for Caputo–Hadamard fractional sub-diffusion equations

In this paper, a second-order scheme with nonuniform time meshes for Caputo–Hadamard fractional sub-diffusion equations with initial singularity is investigated. Firstly, a Taylor-like formula with integral remainder is proposed, which is crucial to studying the discrete convolution kernels and error estimate. Secondly, we come up with an error convolution structure (ECS) analysis for Llog,2−1σ interpolation approximation to the Caputo–Hadamard fractional derivative. The core result in this paper is an ECS bound and a global consistency analysis established at an offset point. By virtue of this result, we obtain a sharp L2-norm error estimate of a second-order Crank–Nicolson-like scheme for Caputo–Hadamard fractional differential equations. Ultimately, an example is presented to show the sharpness of our analysis.

Distorted Acquisition of Dynamic Events Sensed by Frequency-Scanning Fiber-Optic Interrogators and a Mitigation Strategy.

Fiber-optic dynamic interrogators, which use periodic frequency scanning, actually sample a time-varying measurand on a non-uniform time grid. Commonly, however, the sampled values are reported on a uniform time grid, synchronized with the periodic scanning. It is the novel and noteworthy message of this paper that this artificial assignment may give rise to significant distortions in the recovered signal. These distortions increase with both the signal frequency and measurand dynamic range for a given sampling rate and frequency scanning span of the interrogator. They may reach disturbing values in dynamic interrogators, which trade-off scanning speed with scanning span. The paper also calls for manufacturers of such interrogators to report the sampled values along with their instants of acquisition, allowing interpolation algorithms to substantially reduce the distortion. Experimental verification of a simulative analysis includes: (i) a commercial dynamic interrogator of ‘continuous’ FBG fibers that attributes the measurand values to a uniform time grid; as well as (ii) a dynamic Brillouin Optical time Domain (BOTDA) laboratory setup, which provides the sampled measurand values together with the sampling instants. Here, using the available measurand-dependent sampling instants, we demonstrate a significantly cleaner signal recovery using spline interpolation.

Integration of maintenance scheduling and planning for large-scale asset fleets

Large fleets of engineering assets that are subject to ongoing degradation are posing the challenge of how and when to perform maintenance. For a given case study, this paper proposes a formulation for combined scheduling and planning of maintenance actions. A hierarchical approach and a two-stage approach (with either uniform or non-uniform time grid) are considered and compared to each other. The resulting discrete-time linear programming model follows the Resource Task Network framework. Asset deterioration is considered linearly and tackled with an enumerator-based formulation. Advantages of the model are its computational efficiency, scalability, extendability and adaptability. The results indicate that combined maintenance planning and scheduling can be solved in appropriate time and with appropriate accuracy. The decision-support that is delivered helps the choice of the specific maintenance action to perform and proposes when to conduct it. The paper makes a case for the benefits of optimally combining long-term planning and short-term scheduling in industrial-sized problems into one system.

Temporal second-order difference methods for solving multi-term time fractional mixed diffusion and wave equations

This article deals with an establishment and sharp theoretical analysis of a numerical scheme devised for solving the multi-dimensional multi-term time fractional mixed diffusion and wave equations. The governing equation contains both fractional diffusion term and fractional wave term which make the numerical analysis challenging. With the help of the method of order reduction, we convert the time multi-term fractional diffusion and wave terms into the time multi-term fractional integral and diffusion terms respectively, and then develop L2-1σ formula for solving the latter problem. In addition, the formula is used to numerically solve the time distributed-order diffusion and wave equations. The stability and convergence of these numerical schemes are rigorously analyzed by the energy method. The convergence rates are of order two in both time and space. A difference scheme on nonuniform time grids is also constructed for solving the problem with weak regularity at the initial time. Finally, we illustrate our results with some examples.

Plasmas Computed with ATMED CR of the 3rd Non-LTE Code Comparison Workshop Database

In this paper, there are presented some results calculated with ATMED CR of the 3rd Non-LTE Code Comparison Workshop held in December 2003, when this software didn’t exist, having been released in 2017. NLTE population kinetics codes were tested of steady-state cases for C, Al, Ar, Ge, Xe and Au plasmas selected for detailed comparisons. The scope of the meeting consisted of analyzing steady-state dense plasma cases of carbon, low temperature plasmas of aluminium and argon, X-ray laser experiments of germanium also with imposing a Planckian radiation field, medium- and high-Z multicharged ions of hot “experiment-related” plasmas of xenon, using real electron temperature and density parameters inferred from electronic and ionic Thomson scattering spectra and finally plasmas of gold. Being motivated by germanium X-ray laser experiments, the time history of electronic temperature Te and density Ne for a temporal dependent case is provided in Workshop NLTE-3. The calculation with ATMED CR has been carried out to t = 1.975 ns considering the non-uniform time grid along with the corresponding values of Te and Ne presented, being the initial condition LTE at low temperature. 
 The results for plasma properties can be considered as relatively precise and optimal, being checked fundamentally the high sensitivity of calculations to changes in regime, local thermodynamic equilibrium (LTE) or non-LTE (NLTE), electronic and radiation temperatures, electronic density and the percentage of hot electrons. Frequency resolved and mean opacities are also displayed computed with ATMED CR using UTA (Unresolved Transition Array) or Mixed UTA formalisms.

Sharp Error Estimate of the Nonuniform L1 Formula for Linear Reaction-Subdiffusion Equations

Stability and convergence of the L1 formula on nonuniform time grids are studied for solving linear reaction-subdiffusion equations with the Caputo derivative. A discrete fractional Grönwall inequality is developed for the nonuniform L1 formula by introducing a discrete convolution kernel of Riemann--Liouville fractional integral. To simplify the consistency analysis of the nonuniform L1 formula, we bound the local truncation error in a discrete convolution form and consider a global convolution error involving the discrete Riemann--Liouville integral kernel. With the help of discrete fractional Grönwall inequality and global consistency error analysis, a sharp error estimate reflecting the regularity of solution is obtained for a simple L1 scheme. Numerical examples are provided to verify the sharpness of the error analysis.

Multiple and nonuniform time grids in discrete-time MIP models for chemical production scheduling

The modeling of time plays a key role in the formulation of mixed-integer programming (MIP) models for scheduling, production planning, and operational supply chain planning problems. It affects the size of the model, the computational requirements, and the quality of the solution. While the development of smaller continuous-time scheduling models, based on multiple time grids, has received considerable attention, no truly different modeling methods are available for discrete-time models. In this paper, we challenge the long-standing belief that employing a discrete modeling of time requires a common uniform grid. First, we show that multiple grids can actually be employed in discrete-time models. Second, we show that not only unit-specific but also task-specific and material-specific grids can be generated. Third, we present methods to systematically formulate discrete-time multi-grid models that allow different tasks, units, or materials to have their own time grid. We present two different algorithms to find the grid. The first algorithm determines the largest grid spacing that will not eliminate the optimal solution. The second algorithm allows the user to adjust the level of approximation; more approximate grids may have worse solutions, but many fewer binary variables. Importantly, we show that the proposed models have exactly the same types of constraints as models relying on a single uniform grid, which means that the proposed models are tight and that known solution methods can be employed. The proposed methods lead to substantial reductions in the size of the formulations and thus the computational requirements. In addition, they can yield better solutions than formulations that use approximations. We show how to select the different time grids, state the formulation, and present computational results.

Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization. The discretization combines a time stepping on a non-uniform time grid, which generalizes the well-known Störmer–Verlet scheme, with an internal approximation scheme. The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Hölder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces. Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.

Makespan minimization for chemical batch processes using non-uniform time grids

We present a mixed-integer linear programming model for minimizing the makespan of batch processes occurring in the chemical industry. The formulation of the model is based on the starting and processing times of the batches without using an external uniform time grid. A special form for the objective function is chosen and additional redundant constraints are added in order to speed up the solution process. A comparison with the classical uniform time discretization model shows that in many cases the size of the new model as well as its solution time is smaller. Computational results on several instances are reported. The modular structure of the model allows a combination with other continuous-time models for batch processing problems.

A Simple New Continuous-Time Formulation for Short-Term Scheduling of Multipurpose Batch Processes

A new continuous-time formulation for scheduling short-term multipurpose batch processes is presented. The formulation gives rise to a mixed-integer linear programming (MILP) model. The state−task network (STN) representation forms the basis of the proposed approach. A number of event points is prepostulated, which is the same for all tasks in the process. Event times are defined by the ends of task execution, and they are generally different for different tasks of the process, giving rise to a nonuniform time grid. The necessary time monotonicity for single tasks is guaranteed by means of simple duration constraints. Suitable sequencing constraints, applicable to batch tasks involving the same state, are also introduced, so that state balances are properly posed in the context of the nonuniform time grid. The expression of duration and sequencing constraints is greatly simplified by hiding all unit information within the task data. Three benchmark problems are used to illustrate the efficiency and applicability of the new formulation. Results are shown to compare favorably with existing continuous-time formulations in terms of model size and computational effort.