Discrete-time Solution Research Articles

Numerical dynamics for discrete nonlinear damping Korteweg–de Vries equations

We study the numerical scheme of both solution and attractor for the time-space discrete nonlinear damping Korteweg–de Vries (KdV) equation, which is neither conservative nor coercive. First, we establish a new Taylor expansion as well as a global attractor for the KdV lattice system. Second, we prove the unique existence of numerical solution as well as numerical attractor for the discrete-time KdV lattice system via the implicit Euler scheme. Third, we estimate the discretization error and interpolation error between continuous-time and discrete-time solutions, and then establish the upper semi-convergence from numerical attractors to the global attractor as the time-size tends to zero. Fourth, we establish the finitely dimensional approximation of numerical attractors. Finally, we establish the upper bound as well as the lower semi-convergence of numerical attractors with respect to the external force and the damping constant.

Solution of discrete time–cost trade-off problem with adaptive search domain

PurposeExact solution of time–cost trade-off problem (TCTP) by the state-of-the-art meta-heuristic algorithms can be obtained for small- and medium-scale problems, while satisfactory results cannot be obtained for large construction projects. In this study, a hybrid heuristic meta-heuristic algorithm that adapts the search domain is developed to solve the large-scale discrete TCTP more efficiently.Design/methodology/approachMinimum cost slope–based heuristic network analysis algorithm (NAA), which eliminates the unfeasible search domain, is embedded into differential evolution meta-heuristic algorithm. Heuristic NAA narrows the search domain at the initial phase of the optimization. Moreover, activities with float durations higher than the predetermined threshold value are eliminated and then the meta-heuristic algorithm starts and searches the global optimum through the narrowed search space. However, narrowing the search space may increase the probability of obtaining a local optimum. Therefore, adaptive search domain approach is employed to make reintroduction of the eliminated activities to the design variable set possible, which reduces the possibility of converging into local minima.FindingsThe developed algorithm is compared with plain meta-heuristic algorithm with two separate analyses. In the first analysis, both algorithms have the same computational demand, and in the latter analysis, the meta-heuristic algorithm has fivefold computational demand. The tests on case study problems reveal that the developed algorithm presents lower total project costs according to the dependent t-test for paired samples with α = 0.0005.Research limitations/implicationsIn this study, TCTP is solved without considering quality or restrictions on the resources.Originality/valueThe proposed method enables to adapt the number of parameters, that is, the search domain and provides the opportunity of obtaining significant improvements on the meta-heuristic algorithms for other engineering optimization problems, which is the theoretical contribution of this study. The proposed approach reduces the total construction cost of the large-scale projects, which can be the practical benefit of this study.

Stability and convergence of relaxed scalar auxiliary variable schemes for Cahn–Hilliard systems with bounded mass source

Abstract The scalar auxiliary variable (SAV) approach of Shen et al. (2018), which presents a novel way to discretize a large class of gradient flows, has been extended and improved by many authors for general dissipative systems. In this work we consider a Cahn–Hilliard system with mass source that, for image processing and biological applications, may not admit a dissipative structure involving the Ginzburg–Landau energy. Hence, compared to previous works, the stability of SAV-discrete solutions for such systems is not immediate. We establish, with a bounded mass source, stability and convergence of time discrete solutions for a first-order relaxed SAV scheme in the sense of Jiang et al. (2022), and apply our ideas to Cahn–Hilliard systems with mass source appearing in diblock co-polymer phase separation, tumor growth, image inpainting, and segmentation.

A non-standard finite-difference-method for a non-autonomous epidemiological model: analysis, parameter identification and applications.

In this work, we propose a new non-standard finite-difference-method for the numerical solution of the time-continuous non-autonomous susceptible-infected-recovered model. For our time-discrete numerical solution algorithm, we prove preservation of non-negativity and show that the unique time-discrete solution converges linearly towards the time-continuous unique solution. In addition to that, we introduce a parameter identification algorithm for the susceptible-infected-recovered model. Finally, we provide two numerical examples to stress our theoretical findings.

Stochastic Movement Swarm Performing a Coverage Task with Physical Parameters

This paper describes an attempt of implementing physical parameters into a virtual swarm algorithm solution. It defines which physical parameters of the single object need to be known to properly transfer a virtual algorithm into a physical system. Considerations have been based on a stochastic movement swarm performing a coverage task. Time to finish the task and energy consumptions were measured for different numbers of drones in a swarm allowing to designate an optimal size of the swarm. Additional tests for changing variables allowed us to determine their impact on the swarm performance. The presented algorithm is a discrete-time solution and every test is divided into steps. Positions of the drones are calculated only in time corresponding to these steps. Their position is unknown between these steps and the algorithm does not check if the paths of two drones cross between subsequent positions. The lower the time interval, the more precise results, but simulating the test requires more computing power. Further work should consider the smallest possible time intervals or additional feature to check if the paths of the drones do not cross.

Mandel's problem as a benchmark for two-dimensional nonlinear poroelasticity

In this paper, we consider Mandel's problem in the context of nonlinear single-phase poroelasticity, where it is assumed that the fluid is sightly compressible and porosity and permeability are given functions of the volume strain. In the first part of the paper we prove well-posedness of the time-discrete incremental problem by recasting the equations in an abstract form involving a pseudo-monotone operator. Further, we show existence of a Lyapunov functional yielding a global time discrete solution. In the second part, we investigate numerically the behavior of the poroelastic structure. In particular, we verify the assumptions leading to Mandel's solution. We also demonstrate some consequences of the proposed nonlinearities.

Optimal Admission Control for Multiclass Queues with Time-Varying Arrival Rates via State Abstraction

We consider a novel queuing problem where the decision-maker must choose to accept or reject randomly arriving tasks into a no buffer queue which are processed by N identical servers. Each task has a price, which is a positive real number, and a class. Each class of task has a different price distribution, service rate, and arrives according to an inhomogenous Poisson process. The objective is to decide which tasks to accept so that the total price of tasks processed is maximised over a finite horizon. We formulate the problem using a discrete time Markov Decision Process (MDP) with a hybrid state space. We show that the optimal value function has a specific structure, which enables us to solve the hybrid MDP exactly. Moreover, we rigorously prove that as the gap between successive decision epochs grows smaller, the discrete time solution approaches the optimal solution to the original continuous time problem. To improve the scalability of our approach to a greater number of servers and task classes, we present an approximation based on state abstraction. We validate our approach on synthetic data, as well as a real financial fraud data set, which is the motivating application for this work.

Zero-order hold discretization of general state space systems with input delay

Abstract Plethora of applications in physics and engineering are dealing with systems that are subject to input delays. Despite the scientific focus, previous research investigated only state space systems with input delays. Here, we investigate the discretization of generalized state space system with input delay by using the zero-order hold method. Firstly, the solution of a continuous time, generalized state space system with input delay is addressed. By applying the appropriate zero-order hold sampling method, we transform a continuous time linear system into the equivalent discrete time system, while the discrete time solution of the equivalent system is analytically presented. Finally, we use local error metric to estimate the difference between the continuous time and discrete time solutions.

Production scheduling for the reconfigurable modular pharmaceutical manufacturing processes

The novel reconfigurable modular production system is expected to be the next breakthrough in pharmaceutical manufacturing. In these systems, the target product can be manipulated by changing the modules active on a certain production line, but optimal scheduling is a challenging problem because the on-demand operation of the modular system involves sequence-dependent changeover time/cost and competition of limited modules. Based on the characteristics of the modular production system, this work proposes a multi-stage hybrid discrete/continuous-time scheduling framework to fully utilize the advantages of the two formulations. A discrete-time lot-sizing MILP model is first solved to determine the solution structure, and then an information mapping algorithm extracts the necessary information from the discrete-time solution. Finally, a continuous-time LP model is constructed and solved based on the extracted information. The proposed model shows decent results in its application to a reconfigurable modular continuous production system with seven API products.

A cubic nonlinear population growth model for single species: theory, an explicit\u2013implicit solution algorithm and applications

In this paper, we extend existing population growth models and propose a model based on a nonlinear cubic differential equation that reveals itself as a special subclass of Abel differential equations of first kind. We first summarize properties of the time-continuous problem formulation. We state the boundedness, global existence, and uniqueness of solutions for all times. Proofs of these properties are thoroughly given in the Appendix to this paper. Subsequently, we develop an explicit–implicit time-discrete numerical solution algorithm for our time-continuous population growth model and show that many properties of the time-continuous case transfer to our numerical explicit–implicit time-discrete solution scheme. We provide numerical examples to illustrate different behaviors of our proposed model. Furthermore, we compare our explicit–implicit discretization scheme to the classical Eulerian discretization. The latter violates the nonnegativity constraints on population sizes, whereas we prove and illustrate that our explicit–implicit discretization algorithm preserves this constraint. Finally, we describe a parameter estimation approach to apply our algorithm to two different real-world data sets.

A penalized version of the local minimization scheme for rate-independent systems

The letter presents a penalized version of the time-discretization local minimization scheme first proposed by Efendiev and Mielke in 2006 to resolve time discontinuities in rate-independent systems with nonconvex energies. In order to penalize inequality constrains enforcing the local minimality, the Moreau–Yosida approximation is employed. We prove the convergence of time-discrete solutions to functions that are parametrized BV solutions of the time-continuous problem (in an abstract infinite-dimensional setting), provided that the discretization and approximation parameters are chosen appropriately. We test our scheme on a one-dimensional example and find a notable improvement compared with the original version.

Galerkin–collocation approximation in time for the wave equation and its post-processing

We introduce and analyze families of Galerkin–collocation discretization schemes in time for the wave equation. Their conceptual basis is the establishment of a connection between the Galerkin method for the time discretization and the classical collocation methods, with the perspective of achieving the accuracy of the former with reduced computational costs provided by the latter in terms of less complex algebraic systems. Firstly, continuously differentiable in time discrete solutions are studied. Optimal order error estimates are proved. Then, the concept of Galerkin–collocation approximation is extended to twice continuously differentiable in time discrete solutions. A direct link between the two families by a computationally cheap post-processing is presented. A key ingredient of the proposed methods is the application of quadrature rules involving derivatives. The performance properties of the schemes are illustrated by numerical experiments.

Derivation of von Kármán Plate Theory in the Framework of Three-Dimensional Viscoelasticity

We apply a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology to derive a viscoelastic plate model of von K\'arm\'an type. We start from time-discrete solutions to a model of three-dimensional viscoelasticity where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. Combining the derivation of nonlinear plate theory by Friesecke, James and M\"{u}ller, and the abstract theory of gradient flows in metric spaces by Sandier and Serfaty we perform a dimension-reduction from 3D to 2D and identify weak solutions of viscoelastic form of von K\'arm\'an plates.

LQR Design under Stability Constraints

The solution of classic discrete-time, finite-horizon linear quadratic regulator (LQR) problem is well known in literature. By casting the solution to be a static state-feedback, we propose a new method that trades off low LQR objective value with closed-loop stability.

Comparative study of continuously differentiable Galerkin time discretizations for the wave equation

AbstractIn this work we compare numerically two families of space‐time finite element approximation schemes for wave equations. Continuously differentiable in time discrete solutions are provided by the schemes. The first family of schemes is based on a computationally cheap post‐processing of the continuous in space and time finite element approximation of the wave equation. The second family combines the variational approximation in time by finite element techniques with the concepts of collocation methods. Both families show equal order of convergence. The corresponding error constants are reviewed here.

A new method for solving the fission gas diffusion equations with time-varying diffusion coefficient and source term considering recrystallization of fuel grains

• A new time-discrete method is proposed for solving the intergranular gas atom concentration and flux with gas atom resolution and grain recrystallization considered. • Finite element simulation of the irradiation-induced multi-scale thermo-mechanical coupling behavior in a composite fuel pellet is performed. • It is demonstrated that the proposed new method is an efficient method with high accuracy and acceptable computational cost. • The enlarged fission gas diffusion coefficient resulting from the elevated temperature leads to a considerable increase in fuel swelling. • This new method is necessary to be adopted for in-pile behavior analysis of fuel elements under transient and accident conditions. The fission gas swelling of nuclear fuels depends on the fission gas behavior in fuel grains, which will be accelerated by grain recrystallization occurred in some fuels. Under transient and accident conditions, the temperature and fission rate could vary greatly, which will affect the fission gas behavior in fuel grains significantly. It is necessary to develop a new method to solve the fission gas diffusion equations with these effects involved. In this study, for fission gas diffusion equations with time-dependent diffusion coefficient and source term, a new time-discrete method for solving the intergranular gas atom concentration and flux is proposed. With the grain recrystallization considered, the corresponding time-discrete solutions are obtained respectively for the grains in the recrystallized and un-recrystallized areas. The fission gas swelling can be subsequently calculated out. These new mechanistic models are imbedded in the finite element (FE) simulation of the irradiation-induced multi-scale thermo-mechanical coupling behavior in a composite fuel pellet. FE simulation results demonstrate that: (1) the proposed new method is an efficient method with high accuracy and acceptable computational cost; (2) the enlarged fission gas diffusion coefficient will lead to a considerable increase in fuel particle swelling; (3) for composite pellets with high volume fraction of fuel particles, their radial deformation will be heavily affected when the temperature dependence of diffusion coefficient is involved. It can be concluded that the fission gas swelling computation based on this new solution method is necessary to be adopted, when the in-pile behavior simulation is performed for fuel elements under transient and accident conditions.

Unconditional optimal error estimate of the projection/Lagrange-Galerkin finite element method for the Boussinesq equations

This paper provides an unconditional optimal convergence of a fractional-step method for solving the Boussinesq equations. In this method, the convection is treated by the Lagrange-Galerkin technique, whereas the diffusion and the incompressibility are treated by the projection method. There are lots of authors who worked on this method, and some authors gave the error estimate of this method. But, to our best knowledge, the error estimate for this method is under certain time-step restrictions. In this paper, we prove that the methods are stable almost unconditionally, i.e., when τ and h are smaller than a given constant. The basic idea of our analysis is splitting the error function into three terms, one term between the finite element solution and the projection, the other term between the projection and the time-discrete solution, the third term between the time-discrete solution and the exact solution, and giving the error estimates for each term respectively. Then, we obtain the optimal error estimates in L2 and H1-norm for the velocity and L2-norm for the pressure. In order to show the efficiency of our method, some numerical results are presented.

Convergence of discrete and continuous unilateral flows for Ambrosio–Tortorelli energies and application to mechanics

We study the convergence of an alternate minimization scheme for a Ginzburg–Landau phase-field model of fracture. This algorithm is characterized by the lack of irreversibility constraints in the minimization of the phase-field variable; the advantage of this choice, from a computational stand point, is in the efficiency of the numerical implementation. Irreversibility is then recovereda posterioriby a simple pointwise truncation. We exploit a time discretization procedure, with either a one-step or a multi (or infinite)-step alternate minimization algorithm. We prove that the time-discrete solutions converge to a unilateralL2-gradient flow with respect to the phase-field variable, satisfying equilibrium of forces and energy identity. Convergence is proved in the continuous (Sobolev space) setting and in a discrete (finite element) setting, with any stopping criterion for the alternate minimization scheme. Numerical results show that the multi-step scheme is both more accurate and faster. It provides indeed good simulations for a large range of time increments, while the one-step scheme gives comparable results only for very small time increments.

Continuous-Time Locational Marginal Price of Electricity

Modern electricity markets with large penetration of renewable energy resources require fair and accurate pricing methods to elicit generation flexibility and foster competition in electricity markets. This paper proposes the fundamental theory and closed-form formulas for continuous-time locational marginal price (LMP) of electricity, which more accurately integrates the spatio-temporal variations of load and operational constraints of power systems in the electricity price calculation. This paper first formulates the network-constrained generation scheduling and pricing problems as continuous-time optimal control problems using two methods for modeling transmission network, i.e., Theta and generation shift factor (GSF) methods. The continuous-time network-constrained scheduling and pricing problems minimize in their objective functional the total operation cost of power systems over the scheduling horizon subject to generation and transmission constraints. The closed-form continuous-time LMP formulas are derived for each transmission network model, which explicitly include terms that reflect the simultaneous spatio-temporal impacts of transmission flow limits and intertemporal generation ramping constraints in LMP formation. A scalable and computationally efficient function space solution method is proposed that converts the continuous-time problems into mixed integer linear programming problems with finite-dimensional decision space. The proposed solution method enables high-fidelity solution of transmission-constrained scheduling and pricing problems in higher dimensional spaces, while including as a special case the current discrete-time solutions. The proposed models are implemented on a three-bus system and the IEEE reliability test system, where the proposed models showcase more accuracy in reflecting the impacts of fast net-load variations over discrete-time counterparts.

Log-periodicity in piecewise ballistic superdiffusion: Exact results

Small log-periodic oscillations have been observed in many systems and have previously been studied via renormalization-group approaches in the context of critical phenomena [Gluzman and Sornette, Phys. Rev. E 65, 036142 (2002); Derrida and Giacomin, J. Stat. Phys. 154, 286 (2014)]. Here we report their appearance in a random walk model with damaged memory, and we develop an exact discrete-time solution, free from adjustable parameters. Our results shed light on log-periodicity and how it arises. We also discuss continuous-time approaches to the solution along with their limitations and advantages. We show that, as a direct consequence of memory damage, the first moment for the model acquires piecewise ballistic behavior. The piecewise segments are separated by regularly placed singular points. Log-periodicity in this model is seen to be due to memory damage. Remarkably, piecewise ballistic behavior is only observed if one uses the discrete-time solution, because the continuous-time solution does not correctly account for the model's discrete-time dynamics.