Predictor-corrector Iteration Research Articles

A mathematical model for simulating the transmission dynamics of COVID-19 using the Caputo–Fabrizio fractional-order derivative with nonsingular kernel

The emergence of the new coronavirus variant from the coronaviridae family has become a global concern, and all nations, including Bangladesh, are battling to contain the spread of the disease. In this article, we discuss a COVID-19 vaccination model with the Caputo–Fabrizio (CF) fractional order derivative approach to reveal the complicated action in Bangladesh. We determine the existence and uniqueness properties of the outcomes acquired from our proposed model subjected to the Picard–Lindelöf theorem. We perform the stability analysis by using the fixed-point theorem. We utilize Laplace Transform to evaluate the approximate solution of the model. In the numerical simulation, we consider a new approach called the four-step Adams-Bashforth Predictor–Corrector iteration scheme, which simulates that the fractional order provides more precise results. Finally, several numerical results are displayed with the different numbers of the order of the system. We depict the influence of vaccination in the coronavirus model with different parameter values. By analyzing the transmission dynamics of the virus, we bring to light the importance of several doses of vaccination to prevent disease transmission and contagions. We also conducted the sensitivity analysis for the model parameters to evaluate the impact of those parameters on disease outbreaks. The numerical results demonstrate significant information in the Caputo–Fabrizio fractional derivative concept and provide important insights into predicting disease transmission and control policies.

PHƯƠNG PHÁP LẶP SONG SONG RUNGE-KUTTA HAI BƯỚC

In this paper, we introduce the Parallel iteration of two-step Runge-Kutta methods for solving non-stiff initial-value problems for systems of first-order differential equations (ODEs): y′(t) = f(t, y(t)), for use on parallel computers. Starting with an s−stage implicit two-step Runge-Kutta (TSRK) method of order p, we apply the highly parallel predictor-corrector iteration process in P (EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta method that has order p for all m, and that requires s(m+1) right-hand side evaluations per step of which each s evaluation can be computed parallelly. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

Solving primal plasticity increment problems in the time of a single predictor–corrector iteration

The Truncated Nonsmooth Newton Multigrid (TNNMG) method is a well-established method for the solution of strictly convex block-separably nondifferentiable minimization problems. It achieves multigrid-like performance even for non-smooth nonlinear problems, while at the same time being globally convergent and without employing penalty parameters. We show that the algorithm can be applied to the primal problem of classical small-strain elastoplasticity with hardening. Numerical experiments show that the method is considerably faster than classical predictor–corrector methods. Indeed, solving an entire increment problem with TNNMG can take less time than a single predictor–corrector iteration for the same problem. At the same time, memory consumption is reduced considerably, in particular for three-dimensional problems. Since the algorithm does not rely on differentiability of the objective functional, nonsmooth yield laws can be easily incorporated. The method is closely related to a predictor–corrector scheme with a consistent tangent predictor and line search. We explain the algorithm, prove global convergence, and show its efficiency using standard benchmarks from the literature.

きざみ幅制御機能を持つ予測子・修正子型Runge-Kutta-Nystrom 法とその並列計算機への実装

きざみ幅制御機能を持つ予測子・修正子型Runge-Kutta-Nystrom 法とその並列計算機への実装

Convergence Rates of a Class of Predictor-Corrector Iterations for the Nonsymmetric Algebraic Riccati Equation Arising in Transport Theory

AbstractIn this paper, we analyse the convergence rates of several different predictor-corrector iterations for computing the minimal positive solution of the nonsymmetric algebraic Riccati equation arising in transport theory. We have shown theoretically that the new predictor-corrector iteration given in [Numer. Linear Algebra Appl., 21 (2014), pp. 761–780] will converge no faster than the simple predictor-corrector iteration and the nonlinear block Jacobi predictor-corrector iteration. Moreover the last two have the same asymptotic convergence rate with the nonlinear block Gauss-Seidel iteration given in [SIAM J. Sci. Comput., 30 (2008), pp. 804–818]. Preliminary numerical experiments have been reported for the validation of the developed comparison theory.

A comparison of iterative explicit guidance algorithms for space launch vehicles

This paper analyzes and compares the performance of two prominent explicit guidance algorithms: the iterative guidance mode and the powered explicit guidance. We performed a series of numerical simulations of a space launch vehicle model for both nominal and off-nominal conditions. One of our findings is that if we take into account the originally ignored higher-order terms from the guidance parameters of the iterative guidance mode for a long-range flight, the guidance performance can be enhanced to a level comparable to that of the powered explicit guidance. These higher-order terms can be included by employing an iterative predictor–corrector method like the powered explicit guidance. Also we proposed a remedy of preventing relatively earlier divergence of the guidance commands with the predictor–corrector iteration than that of the linear differential corrector approach by making a better initial guess.

Efficient self-consistent quantum transport simulator for quantum devices

We present a self-consistent one-dimensional (1D) quantum transport simulator based on the Contact Block Reduction (CBR) method, aiming for very fast and robust transport simulation of 1D quantum devices. Applying the general CBR approach to 1D open systems results in a set of very simple equations that are derived and given in detail for the first time. The charge self-consistency of the coupled CBR-Poisson equations is achieved by using the predictor-corrector iteration scheme with the optional Anderson acceleration. In addition, we introduce a new way to convert an equilibrium electrostatic barrier potential calculated from an external simulator to an effective doping profile, which is then used by the CBR-Poisson code for transport simulation of the barrier under non-zero biases. The code has been applied to simulate the quantum transport in a double barrier structure and across a tunnel barrier in a silicon double quantum dot. Extremely fast self-consistent 1D simulations of the differential conductance across a tunnel barrier in the quantum dot show better qualitative agreement with experiment than non-self-consistent simulations.

비정상 RANS 법과 중첩격자계를 이용한 횡파중 선박운동 수치해석

The present paper presents the CFD result for a beam wave test case. An ONR tumblehome ship model with bilge keels is used. The beam wave test is for zero forward speed and roll and heave 2DOF with wave slope <TEX>$a_k=0.156$</TEX> and wavelength <TEX>${\lambda}=1.12L_{PP}$</TEX>, with <TEX>$L_{PP}$</TEX> the ship length. The problems is solved numerically with an unsteady Reynolds averaged Navier-Stokes approach. The free surface flow is computed using a single-phase level-set method and the motions in each time step are integrated using a predictor-corrector iteration approach which uses dynamic overset grids moving with relative ship motion. The predicted CFD results for motions and forces are compared with experimental data, showing a reasonable agreement.

Implicit predictor-corrector iteration process for finitely many asymptotically (quasi-)nonexpansive mappings

We study an implicit predictor-corrector iteration process for finitely many asymptotically quasi-nonexpansive self-mappings on a nonempty closed convex subset of a Banach space. We derive a necessary and sufficient condition for the strong convergence of this iteration process to a common fixed point of these mappings. In the case is a uniformly convex Banach space and the mappings are asymptotically nonexpansive, we verify the weak (resp., strong) convergence of this iteration process to a common fixed point of these mappings if Opial's condition is satisfied (resp., one of these mappings is semicompact). Our results improve and extend earlier and recent ones in the literature.

Parallel-Iterated RK-Type PC Methods With Continuous Output Formulas * This work was partly supported by N.R.P.F.S.

This paper investigates parallel predictor-corrector iteration schemes (PC iteration schemes) based on collocation Runge-Kutta corrector methods (RK corrector methods) with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The resulting parallel-iterated RK-type PC methods are also provided with continuous output formulas. The continuous numerical approximations are used for predicting the stage values in the PC iteration processes. In this way, we obtain parallel PC methods with continuous output formulas and high-accurate predictions. Applications of the resulting parallel PC methods to a few widely-used test problems reveal that these new parallel PC methods are much more efficient when compared with the parallel and sequential explicit RK methods from the literature.

Parallel predictor-corrector iteration of pseudo two-step RK methods for nonstiff IVPs

A parallel predictor-corrector (PC) iteration scheme for a general class of pseudo two-step Runge-Kutta methods (PTRK methods) of arbitrarily high order is analyzed for solving first-order nonstiff initial-value problems (IVPs) on parallel computers. Starting with ans-stage pseudo two-step RK method of orderp* withw implicit stages, we apply the highly parallel PC iteration process in P(EC)mE mode. The resulting parallel-iterated pseudo two-step RK method (PIPTRK method) uses an optimal number of processors equal tow. By a number of numerical experiments, we show the superiority of the PIPTRK methods proposed in this paper over both sequential and parallel methods available in the literature.

Fast convergence pirkn-type pc methods with adams-type predictors*

This paper discusses predictor–corrector iteration schemes (PC iteration schemes) based on direct collocatio–based Runge–Kutt–Nyström corrector methods (RKN corrector methods) for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y′′(t) = f(y(t)) Our approach is to regard the well-known parallel-iterated RKN methods (PIRKN methods) as PC iteration processes in which the simple, low-order last step value predictors are replaced with the high-order Adams-type predictors. Moreover, the param-eters of the new direct collocation-based RKN corrector methods are chosen in such a way that the convergence rate of the considered PC iteration processes is optimized. In this way, we obtain parallel PC methods with fast convergence and high-accurate predictions. Application of the resulting parallel PC methods to a few widely-used test problems reveals that the sequential costs are very much reduced when compared with the parallel and sequential explicit RKN methods from the literature.

A class of explicit parallel two-step Runge-Kutta methods

The aim of the present paper is to construct a class of two-step Runge-Kutta methods of arbitrarily high order for application to parallel computers. Starting with ans-stage implicit two-step Runge-Kutta method of orderp withk=p/2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC) m E mode. In this way, we obtain an explicit two-step Runge-Kutta method that has orderp for allm and that requiresk(m+1) right-hand side evaluations per step of which eachk evaluation can be computed in parallel. By a number of numerical experiments we show the superiority of the parallel predictor-corrector methods proposed here over parallel method available in the literature.

Explicit parallel two-step Runge-Kutta-Nyström methods

The aim of this paper is to design a class of two-step Runge-Kutta-Nyström methods of arbitrarily high order for the special second-order equation y″( t ) = f(y( t )), for use on parallel computers. Starting with an s -stage implicit two-step Runge-Kutta-Nyström method of order p with k = p 2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P ( EC ) m E mode. In this way, we obtain an explicit two-step Runge-Kutta-Nyström method that has order p for all m and that requires k ( m + 1) right-hand side evaluations per step of which each k evaluation can be computed in parallel. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

Explicit symmetric Runge-Kutta-Nyström methods for parallel computers

In this paper, we are concerned with parallel predictor-corrector (PC) iteration of Runge-Kutta-Nyström (RKN) methods in P ( EC ) m E mode for integrating initial value problems for the special second-order equation y″( t ) = f(y( t )). We consider symmetric Runge-Kutta-Nyström (SRKN) corrector methods based on direct collocation techniques which optimize the rate of convergence of the PC iteration process. The resulting PISRKN methods (parallel iterated SRKN methods) are shown to be much more efficient when they are compared to the PC iteration process applied to the Gauss-Legendre RKN correctors.

Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems

For the parallel integration of nonstiff initial value problems (IVPs), three main approaches can be distinguished: approaches based on “parallelism across the problem”, on “parallelism across the method” and on “parallelism across the steps”. The first type of parallelism does not require special integration methods and can be exploited within any available IVP solver. The method-parallelism approach received much attention, particularly within the class of explicit Runge-Kutta methods originating from fixed point iteration of implicit Runge-Kutta methods of Gaussian type. The construction and implementation on a parallel machine of such methods is extremely simple. Since the computational work per processor is modest with respect to the number of data to be exchanged between the various processors, this type of parallelism is most suitable for shared memory systems. The required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration and requires the processors to communicate after each full iteration. If the iterations have sufficient computational volume, then the step-parallel approach may be suitable for implementation on distributed memory systems. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The dynamic step-parallel iteration process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to achieve speed-up factors up to 15.

Parallelism across the steps in iterated Runge-Kutta methods for stiff initial value problems

For the parallel integration of stiff initial value problems (IVPs) three main approaches can be distinguished: approaches based on “parallelism across the problem”, on “parallelism across the method” and on “parallelism across the steps”. The first type of parallelism does not require special integration methods can be exploited within any available IVP solver. The methodparallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteraction process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to solve the four-stage Radau IIA corrector used in our experiments within a few effective iterations per step and to achieve speed-up factors up to 10 with respect to the best sequential codes.

Indentation Analysis of Biphasic Articular Cartilage: Nonlinear Phenomena Under Finite Deformation

The nonlinear indentation response of hydrated articular cartilage at physiologically relevant rates of mechanical loading is studied using a two-phase continuum model of the tissue based on the theory of mixtures under finite deformation. The matrix equations corresponding to the governing mixture equations for this nonlinear problem are derived using a total Lagrangian penalty finite element method, and solved using a predictor-corrector iteration within a modified Newton-Raphson scheme. The stress relaxation indentation problem is examined using either a porous (free draining) indenter or solid (impermeable) indenter under fast and slow compression rates. The creep indentation problem is studied using a porous indenter. We examine the finite deformation response and compare with the response obtained using the linear infinitesimal response. Differences between the finite deformation response and the linear response are shown to be significant when the compression rate is fast or when the indenter is impermeable. The finite deformation model has a larger ratio of peak-to-equilibrium reaction force, and higher relaxation rate than the linear model during the early relaxation period, but a similar relaxation time. The finite deformation model predicts a slower creep rate than the linear model, as well as a smaller equilibrium creep displacement. The pressure distribution below the indenter, particularly near the loaded surface is also larger with the finite deformation model.

Parallel block predictor-corrector methods of Runge-Kutta type

In this paper, we construct block predictor-corrector methods using Runge-Kutta correctors. Our approach consists of applying the predictor-corrector method not only at step points, but also at off-step points (block points), so that, in each step, a whole block of approximations to the exact solution is computed. In the next step, these approximations are used to obtain a high-order predictor formula by Lagrange or Hermite interpolation. By choosing the abscissas of the off-step points narrowly spaced, a much more accurately predicted value is obtained than by predictor formulas based on proceding step point values. Since the approximations at the off-step points to be computed in each step can be obtained in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta correctors, the predictor-corrector iteration scheme itself is also highly parallel. Application of these block predictor-corrector methods based on Lagrange-Gauss pairs to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 2 to 11 when compared with the best sequential methods.

Parallel step-by-step methods

In this contribution we survey recent research at CWI on step-by-step methods for solving initial value problems (IVPs) on parallel computers. More general surveys of parallel IVP solvers are given in [5,17,18]. The present paper is organized according to the following sections and subsections: 1. nonstiff problems, 1.1. explicit block methods, 1.2. predictor-corrector iteration, 1.2.1. Runge–Kutta correctors, 1.3. Richardson extrapolation; 2. stiff problems, 2.1. diagonally implicit block methods, 2.2. diagonally implicit iteration, 2.2.1. Runge–Kutta correctors, 2.3. Richardson extrapolation.