Second-order Problems Research Articles

A Parametric Six-Step Method for Second-Order IVPs with Oscillating Solutions

In this paper, we develop an explicit symmetric six-step method for the numerical solution of second-order initial value problems (IVPs) with oscillating solutions. The proposed method is phase-fitted and incorporates a free coefficient as a parameter to optimize its performance. By exploring a wide range of values for this parameter, we computationally determine the periodicity interval. The objective of this procedure is to identify the range of the parameter’s values for which the method remains stable. Based on the output from the periodicity interval analysis, we then aim to define the optimal values for the parameter by numerically solving three initial value problems. The results guided us in identifying these optimal values and confirm the high efficiency of the new method. The method’s efficiency is further validated for the chosen optimal parameter value for specific oscillatory problems, where it is compared with well-known phase-fitted methods.

New implicit time integration schemes for structural dynamics combining high frequency damping and high second order accuracy

The time integration schemes, GA-23 and GA-234, recently proposed by the authors for first order problems, are extended to solve second-order problems in structural dynamics. The resulting methods maintain unconditional stability and user-controlled high-frequency damping. They offer improved accuracy and exhibit less numerical damping in the low-frequency regime, outperforming the well-known generalised-α method. When the high-frequency damping is maximised the new schemes can be cast in the format of backward difference formulae, offering more accurate alternatives to the standard second order formula. The effectiveness of the new time integration schemes is validated through a number of numerical examples, including a linear elastic cantilever beam, a nonlinear spring pendulum, and wave propagation on a string.

PAYOFF-BASED PROBABILISTIC INTERACTION MODEL ON THE EVOLUTION OF COOPERATION IN SPATIAL PUBLIC GOODS GAME

In the spatial public goods game (SPGG), punishment effectively promotes cooperation but often reduces the collective benefits of cooperators and punishers. In order to increase revenue, we propose a probabilistic interaction domain model considering both strategy type and payoff level. In this model, players are divided into two types, successful players with payoffs higher than the payoff threshold and failed players with payoffs lower than the payoff threshold. A successful player is less likely to change the interaction range than a failed player. Through extensive simulations, it is shown to verify that a higher payoff threshold leads to a more pronounced promotion effect on cooperation and corresponds to a higher cooperation return. Moreover, introducing dynamic interaction domain can rapidly remove defectors from the vicinity of cooperative players on regular lattices. Reducing the payoff gap between punishers and cooperators helps mitigate the system’s second-order free-riding problem. Additionally, through analysis of the critical parameters, it is revised that incorporating diversity in interaction structures substantively enhances cooperation level.

Application of first- and second-order adjoint methods to glacial isostatic adjustment incorporating rotational feedbacks

SUMMARY This paper revisits and extends the adjoint theory for glacial isostatic adjustment (GIA) of Crawford et al. (2018). Rotational feedbacks are now incorporated, and the application of the second-order adjoint method is described for the first time. The first-order adjoint method provides an efficient means for computing sensitivity kernels for a chosen objective functional, while the second-order adjoint method provides second-derivative information in the form of Hessian kernels. These latter kernels are required by efficient Newton-type optimization schemes and within methods for quantifying uncertainty for non-linear inverse problems. Most importantly, the entire theory has been reformulated so as to simplify its implementation by others within the GIA community. In particular, the rate-formulation for the GIA forward problem introduced by Crawford et al. (2018) has been replaced with the conventional equations for modelling GIA in laterally heterogeneous earth models. The implementation of the first- and second-order adjoint problems should be relatively easy within both existing and new GIA codes, with only the inclusions of more general force terms being required.

Smoothing penalty approach for solving second-order cone complementarity problems

Smoothing penalty approach for solving second-order cone complementarity problems

A reliable ensemble forecasting modeling approach for complex time series with distributionally robust optimization

Accurate ensemble forecast results provide strong support for management decisions. While the existing ensemble forecasting model ignores the requirement of directional accuracy making its prediction direction is usually error-prone. To address this issue, we develop a convex directional prediction error measure and subsequently construct a reliable ensemble forecasting model that minimizes the horizontal prediction error with the bounded directional prediction error. Mathematically, we provided the proper range of the required directional error level. Furthermore, we find the classical ensemble forecasting model is a special case of our study when the directional error level is larger than the upper bound we gave in this study. To deal with the possible deterioration of the individual model over time, we also considered its worst possible prediction performance by introducing the distributionally robust optimization (DRO) technique into the proposed reliable ensemble forecasting model. Technically, we showed that the DRO-based reliable ensemble forecasting model is convex and can be reformulated into a second-order cone problem which can be readily solved by off-the-shelf solvers. Finally, the effectiveness of the proposed reliable ensemble forecasting model and the DRO-based reliable ensemble forecasting model were validated on three different datasets, e.g., the exchange rate, the oil price, and the country risk index. To sum up, we construct a reliable ensemble forecasting model to simultaneously control the horizontal prediction error and directional prediction error of ensemble forecasting, and thus enhance the reliability of ensemble forecasting.

A variational-based non-smooth contact dynamics approach for the seismic analysis of historical masonry structures

A variational formulation of the non-smooth contact dynamics method is proposed to address the dynamic response of historical masonry structures modeled as systems of 3D rigid blocks and subjected to ground excitation. Upon assuming a unilateral-frictional contact law between the blocks, the equations of motions are formulated in a time-discrete impulse theorem format in the unknown block velocities and contact impulses. The variational structure of the problem to be solved at each time step is proven. On that basis, the numerical method requires at each time step to perform a collision detection that identifies antagonist contact points based on the given structural configuration, to solve a second-order conic programming problem that outputs block velocities and contact impulses, and to update the structural configuration for the solution to advance in time. As a merit of the formulation, large-scale problems can be robustly and efficiently addressed thanks to the convex setting of the time-step optimization problem. Numerical results are presented to test the computational performances of the proposed approach. Benchmark problems provide numerical evidence that the formulation is consistent with event-driven solutions based on the classical Housner impact model. The dynamic response, failure domains, and fragility functions of real-size masonry structures are then explored under ground impulse or earthquake excitation. The obtained results prove the reliability of the present computational method for the dynamic analysis and seismic assessment of historical masonry constructions of engineering interest.

Byzantine-Resilient Second-Order Consensus in Networked Systems.

This article studies the second-order consensus problem in networked systems containing the so-called Byzantine misbehaving nodes when only an upper bound on either the local or the total number of misbehaving nodes is known. The existing results on this subject are limited to malicious/faulty model of misbehavior. Moreover, existing results consider consensus among normal nodes in only one of the two states, with the other state converging to either zero or a predefined value. In this article, a distributed control algorithm capable of withstanding both locally bounded and totally bounded Byzantine misbehavior is proposed. When employing the proposed algorithm, the normal nodes use a combination of the two relative state values obtained from their neighboring nodes to decide which neighbors should be ignored. By introducing an underlying virtual network, conditions on the robustness of the communication network topology for consensus on both states are established. Numerical simulation results are presented to illustrate the effectiveness of the proposed control algorithm.

Analytical solution to the simultaneous Michaelis-Menten and second-order kinetics problem

Analytical solution to the simultaneous Michaelis-Menten and second-order kinetics problem

Second-order Karush/Kuhn–Tucker conditions and duality for constrained multiobjective optimization problems

In this paper, we investigate the second-order strong Karush/Kuhn–Tucker conditions and duality of a constrained multiobjective optimization problem (CMOP). Exploiting a second-order regularization condition, we obtain second-order strong KKT necessary conditions of Borwein-properly efficient solution of CMOP without convexity assumptions. Further, second-order sufficient conditions for the second-order KKT point to be an efficient solution of CMOP are derived under the generalized second-order convexity assumptions. Finally, we establish duality results between CMOP and its second-order Wolfe-type dual problem.

A compact combination of second-kind Chebyshev polynomials for Robin boundary value problems and Bratu-type equations

AbstractThis paper presents a new algorithm for resolving linear and non-linear second-order Robin boundary value problems (BVPS) and the Bratu-type equations in one and two dimensions using spectral approaches. Basis functions according to second-kind shifted and modified shifted Chebyshev polynomials that comply with the Robin conditions are created. It has produced operational matrices for its derivatives. The provided solutions are the result of applying the collocation and tau approaches. These methods convert the problem dictated by its boundary conditions into a system of linear or non-linear algebraic equations that may be solved using any suitable numerical solver. Convergence analysis has been provided and it accords with the numerical results. Six numerical problems are provided to investigate and demonstrate the practical utility of the suggested method. The current results show that our method outperforms the previous methods in terms of accuracy which are presented in tables and figures.

Existence of Homoclinic Solutions for a Class of Nonlinear Second-order Problems

Existence of Homoclinic Solutions for a Class of Nonlinear Second-order Problems

An Effective Matrix Technique for the Numerical Solution of Second Order Differential Equations

In this paper, an effective technique for solving differential equations with initial conditions is presented. The method is based on the use of the Legendre matrix of derivatives defined on the close interval [-1,1]. Properties of the polynomial are outlined and further used to obtain the matrix of derivative which was used in transforming the differential equation into systems of linear and nonlinear algebraic equations. The systems of these algebraic equations were then solved using Gaussian elimination method to determine the unknown parameters required for approximating the solution of the differential equation. The advantage of this technique over other methods is that, it has less computational manipulations and complexities and also its availability for application on both linear and nonlinear second-order initial value problems is impressive. Other advantage of the algorithm is that high accurate approximate solutions are achieved by using a greater number of terms of the Legendre polynomial and once the operational matrix is obtained, it can be used to solve differential equations of higher order by introducing just a little manipulation on the operational matrix. Some existing sample problems from literature were solved and the results were compared to show the validity, simplicity and applicability of the proposed method. The results obtained validate the simplicity and applicability of the method and it also reveals that the method perform better than most existing methods.

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

Evolutionary dynamics of any multiplayer game on regular graphs

Multiplayer games on graphs are at the heart of theoretical descriptions of key evolutionary processes that govern vital social and natural systems. However, a comprehensive theoretical framework for solving multiplayer games with an arbitrary number of strategies on graphs is still missing. Here, we solve this by drawing an analogy with the Balls-and-Boxes problem, based on which we show that the local configuration of multiplayer games on graphs is equivalent to distributing k identical co-players among n distinct strategies. We use this to derive the replicator equation for any n-strategy multiplayer game under weak selection, which can be solved in polynomial time. As an example, we revisit the second-order free-riding problem, where costly punishment cannot truly resolve social dilemmas in a well-mixed population. Yet, in structured populations, we derive an accurate threshold for the punishment strength, beyond which punishment can either lead to the extinction of defection or transform the system into a rock-paper-scissors-like cycle. The analytical solution also qualitatively agrees with the phase diagrams that were previously obtained for non-marginal selection strengths. Our framework thus allows an exploration of any multi-strategy multiplayer game on regular graphs.

Improving interdependent urban power and gas distribution systems resilience through optimal scheduling of mobile emergency supply and repair resources

Urban electric power and natural gas systems, serving as the most critical facilities that provide fundamental support to people's daily life, have been increasingly interdependent in their operation over the past decades. This growing interdependency presents great challenges and demands for enhancing the resilience of the interdependent systems, particularly driven by the recent cold weather events. In this paper, we propose a comprehensive strategy for scheduling various types of emergency resources, including mobile and stationary generators, repair crews, and fuel tankers, with the goal of enhancing the resilience of interdependent power and gas distribution systems in the aftermath of disasters. In particular, the strategy incorporates the routing of mobile resources to orchestrate their dynamic moving behaviors. Fuel supply issue for mobile and stationary distributed generators is also addressed through modeling the fuel exchange processes among generators, fuel tankers, and locations. Furthermore, we present a model of repair process to fully capture the real-world repair activities, allowing varying numbers of repair crews to work in a cooperative way. The proposed strategy is formulated into a tractable mixed-integer second-order cone problem. Finally, case studies are carried out to demonstrate the effectiveness of the strategy in enhancing the resilience of the interdependent power and gas distribution systems.

Hp-version C1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

<i>hp</i>-version <i>C</i>1-continuous Petrov–Galerkin method for nonlinear second-order initial value problems with application to wave equations

Generating linear, semidefinite, and second-order cone optimization problems for numerical experiments

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems.

Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is &amp;lt;i&amp;gt;O(h&amp;lt;sup&amp;gt;p+2&amp;lt;/sup&amp;gt;)&amp;lt;/i&amp;gt; superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations

Postprocessing Techniques of High-Order Galerkin Approximations to Nonlinear Second-Order Initial Value Problems with Applications to Wave Equations