Rosenbrock Method Research Articles

A modified Susceptible-Infected-Recovered epidemiological model

"Objectives This paper proposes an infectious disease model incorporating two new model compartments, hospitalization, and intensive care unit. Methods The model dynamics are analyzed using the local and global stability theory of nonlinear systems of ordinary differential equations. For the numerical simulations, we used the Rosenbrock method for stiff initial value problems. We obtained numerical simulations using MAPLE software. The returned MAPLE procedure was called only for points inside the range on which the method evaluated the numerical solution of the system with specified initial conditions. Results We proposed a new model to describe the dynamics of microparasitic infections. Numerical simulations revealed that the proposed model fitted with the expected behaviour of mi- croparasitic infections with ”acute epidemicity.” The numerical simulations showed consistency in the behaviour of the system. Conclusions The model proposed has ”robust” dynamics, supported by the global stability of its endemic state and the consistency of the numerical simulations regarding the model’s timeevolution behaviour. The introduction of the hospitalization and intensive care unit compartments in the proposed model revealed that it is essential to consider such policies in the case of ”acuteepidemicity” of microparasitic infections."

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix $\varphi$-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods.

Efficient adaptive step size control for exponential integrators

Traditional step size controllers make the tacit assumption that the cost of a time step is independent of the step size. This is reasonable for explicit and implicit integrators that use direct solvers. In the context of exponential integrators, however, an iterative approach, such as Krylov methods or polynomial interpolation, to compute the action of the required matrix functions is usually employed. In this case, the assumption of constant cost is not valid. This is, in particular, a problem for higher-order exponential integrators, which are able to take relatively large time steps based on accuracy considerations. In this paper, we consider an adaptive step size controller for exponential Rosenbrock methods that determines the step size based on the premise of minimizing computational cost. The largest allowed step size, given by accuracy considerations, merely acts as a constraint. We test this approach on a range of nonlinear partial differential equations. Our results show significant improvements (up to a factor of 4 reduction in the computational cost) over the traditional step size controller for a wide range of tolerances.

A brief history of Accident Chernobyl: Simulation of the influence of Neutron Absorbing Poisons and temperature feedback effects by Point Kinetics Equations

In this paper, the solution of the Neutron Point Kinetics model is presented, adding the effects of temperature and absorbers poisons within a historical and technical context to simulate the preliminary characteristics of the Chernobyl accident. The Point Kinetics model was able to extract physical information consistent with what was expected to predict the reactor situation until the accident. It was also possible to verify, given the results, that the Rosenbrock method was able to overcome the degree of stiffness of the ODE system, besides solving a non-linear problem. Thus, this study has contributed to highlighting the importance of temperature effects and especially absorbers poisons in the final power behavior, extremely relevant for decision making in the operation and safety of a nuclear power plant.

A Class of Adaptive Exponentially Fitted Rosenbrock Methods with Variable Coefficients for Symmetric Systems

In several important scientific fields, the efficient numerical solution of symmetric systems of ordinary differential equations, which are usually characterized by oscillation and periodicity, has become an open problem of interest. In this paper, we construct a class of embedded exponentially fitted Rosenbrock methods with variable coefficients and adaptive step size, which can achieve third order convergence. This kind of method is developed by performing the exponentially fitted technique for the two-stage Rosenbrock methods, and combining the embedded methods to estimate the frequency. By using Richardson extrapolation, we determine the step size control strategy to make the step size adaptive. Numerical experiments are given to verify the validity and efficiency of our methods.

On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations

In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments.

Nonlinear Mathematical Model for Dynamic Buckling of Stiffened Orthotropic Shell Panels

This presents a mathematical model proposed by the author for description of deformation process of a shell structure under the action of a load that depends linearly on time. Material orthotropy, geometric nonlinearity, transverse shifts are taken into account. A distinctive feature of the model is the use of a refined discrete method for taking into account stiffeners, proposed by the author earlier. Prior to this, the method was used only in static or isotropic dynamic problems. It is proposed to add correction normalizing factors, which makes it possible to obtain the most accurate values of critical loads. The methodology of the calculation algorithm under dynamic loading is based on the Kantorovich method and the Rosenbrock method, which allows solving rigid ordinary differential equation (ODE) systems. New numerical results for cylindrical panels are obtained. The influence of the number of stiffening elements on the values of the critical load is shown. A comparison with the classical discrete method of taking into account stiffeners is carried out. For the problems considered in this paper, the phase portraits of the system are shown.

Application of MGA and EGA algorithms on large-scale linear systems of ordinary differential equations

In this paper, we propose two new approaches to solve the large linear system of ordinary differential equations (LSODE), which are based on the projection technique on the extended global Krylov (EGKS) or global Krylov (GKS) subspaces if A is not invertible. The first approach (EGA-expm or MGA-expm if A is not invertible) is based on the projection of matrix exponential function (expm), the second approach (EGA-ROS2, EGA-BDF2 or MGA-ROS2, MGA-BDF2 if A is not invertible) is based on a projection of the initial problem onto an EGKS (or GKS if A is not invertible) to get a new LSODE of small-size then solve it with the 2-stage Rosenbrock method (ROS2) or the backward differentiation formula (BDF2) method and the 2-stage obtained solution is used to have a low-rank (LR) approximate solution of the initial problem. Moreover, we will present the expression of the residual and the error norms. Finally, some attractive and interesting examples of specific problems are presented to confirm the strength of the new approaches.

An efficient rotational direction heap-based optimization with orthogonal structure for medical diagnosis

The heap-based optimizer (HBO) is an optimization method proposed in recent years that may face local stagnation problems and show slow convergence speed due to the lack of detailed analysis of optimal solutions and a comprehensive search. Therefore, to mitigate these drawbacks and strengthen the performance of the algorithm in the field of medical diagnosis, a new MGOHBO method is proposed by introducing the modified Rosenbrock's rotational direction method (MRM), an operator from the grey wolf optimizer (GWM), and an orthogonal learning strategy (OL). The MGOHBO is compared with eleven famous and improved optimizers on IEEE CEC 2017. The results on benchmark functions indicate that the boosted MGOHBO has several significant advantages in terms of convergence accuracy and speed of the process. Additionally, this article analyzed the diversity and balance of MGOHBO in detail. Finally, the proposed MGOHBO algorithm is utilized to optimize the kernel extreme learning machines (KELM), and a new MGOHBO-KELM is proposed. To validate the performance of MGOHBO-KELM, seven disease diagnostic questions were introduced for testing in this work. In contrast to advanced models such as HBO-KELM and BP, it can be concluded that the MGOHBO-KELM model can achieve optimal results, which also proves that it has practical significance in solving medical diagnosis problems.

Exponential Multistep Methods for Stiff Delay Differential Equations

Stiff delay differential equations are frequently utilized in practice, but their numerical simulations are difficult due to the complicated interaction between the stiff and delay terms. At the moment, only a few low-order algorithms offer acceptable convergent and stable features. Exponential integrators are a type of efficient numerical approach for stiff problems that can eliminate the influence of stiffness on the scheme by precisely dealing with the stiff term. This study is concerned with two exponential multistep methods of Adams type for stiff delay differential equations. For semilinear delay differential equations, applying the linear multistep method directly to the integral form of the equation yields the exponential multistep method. It is shown that the proposed k-step method is stiffly convergent of order k. On the other hand, we can follow the strategy of the Rosenbrock method to linearize the equation along the numerical solution in each step. The so-called exponential Rosenbrock multistep method is constructed by applying the exponential multistep method to the transformed form of the semilinear delay differential equation. This method can be easily extended to nonlinear delay differential equations. The main contribution of this study is that we show that the k-step exponential Rosenbrock multistep method is stiffly convergent of order k+1 within the framework of a strongly continuous semigroup on Banach space. As a result, the methods developed in this study may be utilized to solve abstract stiff delay differential equations and can be served as time matching methods for delay partial differential equations. Numerical experiments are presented to demonstrate the theoretical results.

Numerical event location techniques in discontinuous differential algebraic equations

In this paper, we consider numerical methods for the location of events of differential algebraic equations of index one. These events correspond to cross a discontinuity surface, beyond which another differential algebraic equation holds. Convergence theorems of the numerical event time and event point to the true event time and event point are given. It is proved that, for integrations by semi-implicit methods or Rosenbrock methods, the order of convergence of the numerical event location is the order of convergence of the continuous extension and, for integration by implicit Runge-Kutta methods, the order of convergence is the order of convergence of the implicit RK method.

ОСОБЕННОСТИ ВТОРИЧНОГО ЗАГРЯЗНЕНИЯ ВОЗДУХА ПРИ ИЗМЕНЕНИИ ПЛОТНОСТИ ПОТОКА ТРАНСПОРТА

The pollution of the urban atmosphere in the area of the direct action of the motorway is considered. A certified environmental impact assessment methodology was used as the master. With its help, the masses of substances – that are primary pollutants – emitted per unit of time are calculated depending on the intensity of the traffic flow. Further, to calculate their concentrations, the dispersion is calculated. In order to determine the concentration of the secondary pollutant, ozone, a mathematical model is proposed. It is a stiff system of differential equations that describes the kinetics of the process of chemical transformation of primary pollutants. To solve it, the Rosenbrock method was applied. The influence of the traffic flow density on the ozone content near the highway is being studied

Multirate linearly-implicit GARK schemes

Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

ALGORITHM OF CONTROLLING AN ADAPTIVE HYDRAULIC CIRCUIT FOR MOBILE MACHINES

Hydraulic systems based on adjustable pumps, proportional electrohydraulic equipment and controllers are used in mobile machines. The authors propose a new scheme of the hydraulic system for mobile machines, which provides the auger drilling operation. A number of studies have shown that a certain ratio should be maintained between the frequency of auger rotation and its feed during operation, where the productivity of soil disruption should not exceed the productivity of transporting loose soil from the drilling zone. Ensuring the required ratio between the speed of the auger rotation and its feed is implemented by a controller that works according to a certain algorithm. A nonlinear mathematical model of the hydraulic system was developed to create the algorithm for controller operation and setting. The equations of the mathematical model are solved in the MATLAB-Simulink environment by the Rosenbrock method. As a result of solving the equations for the mathematical model, the dependences of variables describing the state of the hydraulic system on time are obtained. The values of the controller settings are determined at which the hydraulic system works steadily, the error of flow rate stabilization, the time for pressure adjustment and readjustment does not exceed the allowable values. The algorithm for controlling the auger feed value is formed. This algorithm provides the necessary ratio between the auger feed and speed, as well as reducing the feed rate in the case of soil hardness increases. This creates the conditions for uninterrupted pit drilling at full depth and protection of the hydraulic system from overload.

On Adaptive Patankar Runge–Kutta methods

AbstractWe apply Patankar Runge–Kutta methods to y′ = M(y)y and focus on the case where M(y) is a graph Laplacian as the resulting scheme will preserve positivity and total mass. The second order Patankar Heun method is tested using four test problems (stiff and non‐stiff) cast into this form. The local error is estimated and the step size is chosen adaptively. Concerning accuracy and efficiency, the results are comparable to those obtained with a traditional L‐stable, second order Rosenbrock method.

High‐order implicit time‐stepping with high‐order central essentially‐non‐oscillatory methods for unsteady three‐dimensional computational fluid dynamics simulations

AbstractThis article develops high‐order implicit time‐stepping methods combined with the fourth‐order central essentially‐non‐oscillatory (CENO) scheme for stiff three‐dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three‐dimensional multiblock body‐fitted grids with hexahedral cells. Several implicit time integration methods of third‐ and fourth‐order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge‐Kutta (ESDIRK4), and Rosenbrock‐type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian‐free inexact Newton–Krylov method with additive Schwarz preconditioning using block‐based incomplete LU decomposition. The performance of the high‐order implicit time‐stepping methods on smooth and stiff problems is compared with a standard fourth‐order explicit Runge‐Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth‐order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth‐order CENO method. When applied to the problem of vortex shedding governed by the Navier–Stokes equations, an A‐stable ESDIRK4 scheme proved to be the more robust and accurate implicit time‐marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high‐order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high‐order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high‐order accuracy, and it also represents the first to establish high‐order implicit time integration for high‐order CENO in space.

Efficient Time Integration of Nonlinear Partial Differential Equations by Means of Rosenbrock Methods

We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitable choice of boundary values for the internal stages. The main difference from the linear case comes from the difficulty to calculate those boundary values exactly in terms of data. In any case, the implementation is cheap and simple since, at each stage, just some additional terms concerning those boundary values and not the whole grid must be added to what would be the standard method of lines.

Solving variable-coefficient burnup equations by exponential Rosenbrock methods

Burnup calculation is an essential topic in reactor physics analysis, which is based on the neutronic-depletion coupling calculation. High order neutronic-depletion coupling methods have been demonstrated much superior to the traditional coupling methods in terms of accuracy, but introduce the variable-coefficient burnup equations. The straightforward and general method for solving variable-coefficient burnup equations is to discretize the burnup step into several substeps and then approximate variable-coefficient equations by a sequence of constant-coefficient ones. However, substep sizes must be rather small to obtain satisfactory accuracy, which causes numerous computational cost. In this work, a class of one-step methods called exponential Rosenbrock methods is proposed to solve variable-coefficient burnup equations. The major part of the implementation of exponential Rosenbrock methods is computing the matrix exponentials and related φ functions, so the matrix exponential methods for constant-coefficient burnup equations can be applied to exponential Rosenbrock methods directly. Moreover, each exponential Rosenbrock method has a corresponding embedded error estimator for adaptive time-stepping. A 4th-order exponential Rosenbrock method called EXPRB43 is used in this work, and two EXPRB43-based predictor–corrector coupling methods called EXPRB43-CE/LI and EXPRB43-LE/QI are further established by replacing the time step discretization method used in the conventional CE/LI and LE/QI with EXPRB43. To investigate the correctness and performance of EXPRB43 for solving variable-coefficient burnup equations, three artificial test Cases are first constructed by combining three different time-dependent fluxes and a specific cross-section library. The numerical results show EXPRB43 can improve the computational efficiency by 2 to 3 orders of magnitude compared to the straightforward method, and this novel method could easily obtain more accurate and reliable solutions by using an adaptive stepsize controller. Then, a test case for depleting a 2D UO2-Gd fuel pin is investigated to further demonstrate the performance of EXPRB43 in realistic neutronic-depletion coupling problems. By comparing to the conventional CE/LI and LE/QI, EXPRB43-based coupling methods can obtain slightly more accurate solutions especially for several important fission products, such as 155Gd, 157Gd, 135Xe and 149Sm. Although the overall computational errors dominated by the errors of polynomial approximation of the time dependent burnup matrix cover up the improvement of EXPRB43 to the naive substep method, EXPRB43-based coupling methods can be alternative to the conventional ones.

Enhanced Harris hawks optimization with multi-strategy for global optimization tasks

Harris Hawks Optimization (HHO) algorithm is a newly proposed meta-heuristic optimization algorithm that simulates the hunting process of the Harris hawks. It has the characteristics of fewer adjustment parameters and a strong optimization effect, resulting in strong competitiveness in similar optimization algorithms. However, HHO is prone to premature convergence and low convergence accuracy when dealing with specific complex optimization problems. Therefore, our work integrates two novel strategies into the standard HHO to gain enhanced exploration and exploitation capabilities. Specifically, our work firstly proposed an exploration strategy based on logarithmic spiral and opposition-based learning to improve the exploration ability of HHO. Secondly, the local search technique for Rosenbrock Method (RM) is modified to dynamically fuse into the standard HHO to enhance the algorithm’s local search capability and improve the convergence accuracy. The novel meta-heuristic algorithm proposed in this paper is called RLHHO. Finally, to validate the algorithm’s effectiveness, the proposed RLHHO algorithm is fully performance tested with eight other traditional meta-heuristic optimization algorithms on 23 benchmark functions and 30 IEEE CEC2014 test functions. Besides, another six advanced meta-heuristics algorithms are also compared in the 30 CEC’2014 test functions. The experimental results show that RLHHO performs significantly better than HHO as well as other traditional and advanced meta-heuristic algorithms in most of the test functions. To test the scalability of RLHHO in complex real-world problems, it was used to optimize the solution of three constrained real-world engineering problems, and the experimental results show that RLHHO’s powerful performance can be used as an effective tool for solving constrained engineering problems. Also, an effective hybrid model of kernel extreme learning machine is developed on the basis of RLHHO to cope with bankruptcy prediction problem. The experimental results show that this hybrid model is highly competitive with other mainstream classifiers regarding stability and prediction accuracy. The supplementary info and answers to possiblequeries will be publicly available at https://www.researchgate.net/profile/Chenyang_Li39/research.

Particle Swarm Optimization Combined with Inertia-Free Velocity and Direction Search

The particle swarm optimization algorithm (PSO) is a widely used swarm-based natural inspired optimization algorithm. However, it suffers search stagnation from being trapped into a sub-optimal solution in an optimization problem. This paper proposes a novel hybrid algorithm (SDPSO) to improve its performance on local searches. The algorithm merges two strategies, the static exploitation (SE, a velocity updating strategy considering inertia-free velocity), and the direction search (DS) of Rosenbrock method, into the original PSO. With this hybrid, on the one hand, extensive exploration is still maintained by PSO; on the other hand, the SE is responsible for locating a small region, and then the DS further intensifies the search. The SDPSO algorithm was implemented and tested on unconstrained benchmark problems (CEC2014) and some constrained engineering design problems. The performance of SDPSO is compared with that of other optimization algorithms, and the results show that SDPSO has a competitive performance.