Embedded Runge-Kutta Methods Research Articles

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

Stability of step size control based on a posteriori error estimates

A posteriori error estimates based on residuals can be used for reliable error control of numerical methods. Here, we consider them in the context of ordinary differential equations and Runge-Kutta methods. In particular, we take the approach of Dedner & Giesselmann (2016) and investigate it when used to select the time step size. We focus on step size control stability when combined with explicit Runge-Kutta methods and demonstrate that a standard I controller is unstable while more advanced PI and PID controllers can be designed to be stable. We compare the stability properties of residual-based estimators and classical error estimators based on an embedded Runge-Kutta method both analytically and in numerical experiments.

Projective integration methods in the Runge–Kutta framework and the extension to adaptivity in time

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge–Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge–Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge–Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge–Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.

Error Analysis Using Three and Four Stage Eighth Order Embedded Runge-Kutta Method for Sixth Order Ordinary Differential Equation vvi(u)=f(u,v,v',v'',v''',viv)

The present paper aims at providing an insight to embedded Runge-Kutta sixth order (RKSD) ordinary differential equation method for solving the initial value problem of order six of type vvi(u) = f(u, v, v', v'',v''',viv). The concept of order conditions for the three and four stages up to the eighth and ninth orders, respectively, is designed and evaluated; furthermore, the zero-stability of the proposed method is proved. Comparisons are made between these orders with the help of a mathematical example, and global and local truncated error norms are evaluated.

Modeling the effect of application-specific program transformations on energy and performance improvements of parallel ODE solvers

Ordinary differential equations (ODEs) are important for modelling many problems from science and engineering and efficient ODE solvers are required, for example when solving time-dependent partial differential equations (PDEs) with the method of lines. Since an ODE solver may perform a large number of iteration steps, the execution time for solving an ODE problem might be quite large. Thus, a reduction of the execution time is desirable and should affect each iteration step of the simulation. Programming techniques to reduce the execution time of ODE solver are parallelism and modification of the memory access structure such that the memory access time decreases. In this article, we investigate multithreaded solution methods for ODEs with different memory access behavior and their influence on the performance. Additionally the energy consumption is considered. The parallelism is implemented as shared memory program for multicore processors. The memory access behavior is investigated using different program variants which result from application-specific program transformations changing the memory access order while guaranteeing the numerical correctness. For the investigation of the performance, experimental data have been gathered on five different recent multicore processors. Additionally, an analytical power and energy model for modeling the performance and energy consumption is introduced. As ODE solver, the popular embedded Runge-Kutta methods with error correction is used. The simulation problems are two different ODEs resulting from discretized PDEs. The experimental data give insight into the quite diverse performance behavior of the ODE solver variants solving the same problem on different platforms.

Based on Particle Swarm Optimization Algorithm for Finding Ideal and Effective Solution of Dynamic Batch Reactor Multi-objective Optimization

Chemical production process integration is higher and higher, the operation is more and more complex. Based on the dynamic simulation of the process, analysis automation engineering and obtain decision, is beneficial to improve the production efficiency, also ensure safety. The mathematical model is a multi-objective dynamic optimization problem, and it is difficult to solve the multi-objective and it involves the numerical solution of differential system. The particle swarm algorithm strategy is given in this paper for the problem of a numerical solution. Penalty term was added to the PSO algorithm is put forward, as well as the local extremum and global extremum was used to adjust further, so that PSO algorithm is suitable for finding the ideal efficient solutions for multi-objective optimization problem, at the same time in the process of evolution, embedded Runge-Kutta method, the algorithm could been used for dynamic multi-objective optimization problems.

Prediction of Pyroshock-Reduced Separation Nut Behaviors

In this work, an efficient model is proposed to predict the complicated coupling behavior of a pyroshock-reduced separation nut, which has two variable-volume chambers connected by the vent hole. The model considers the effect of mutual interactions between combustion products and internal moving parts of the separation nut. To accomplish this, governing equations for combustion modeling are systematically derived, assuming that the flowfield within each chamber is homogeneous. For internal moving parts of the separation nut, the equations of motion are constructed. The equations are used to construct a coupled model. The behavior of the model from solving the system of differential equations is obtained, based on the Dormand–Prince embedded Runge–Kutta method. To validate the model, its behavior is compared with the test results of an actual prototype. Furthermore, modeling effects of the condensed phase product and initial air are investigated, and their significance in combustion modeling is identified. Parametric studies are then carried out to investigate the effects of the design parameters on the separation behavior. The results indicate that the model could be used efficiently in the design of separation nuts.

On nonlinear vibrations of micropipes conveying fluid

This work investigates nonlinear size-dependent resonant characteristics of fluid-conveying extensible micropipes subjected to a harmonic load. The nonlinear governing equation and corresponding boundary conditions of system are developed on the basis of modified couple stress continuum theory in conjunction with Euler–Bernoulli beam theory and von Kármán's geometric nonlinearity. Galerkin technique is employed to discretize the integro-partial-differential governing equation into a set of second-order nonlinear ordinary differential equations with coupled terms. After that, an embedded Runge–Kutta method is utilized to solve numerically the resultant equations. The nonlinear size-dependent primary resonant characteristics of a simply supported micropipe conveying fluid in subcritical domain are examined via depicting frequency-response and force-response curves. The influences of different parameters i.e., flexural rigidity ratio which represent the effect of size-dependency, slenderness ratio, and dimensionless mean flow velocity on the nonlinear size-dependent forced vibration characteristics of system are examined.

A comparative analysis of local meshless formulation for multi-asset option models

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

SPIP: A computer program implementing the Interaction Picture method for simulation of light-wave propagation in optical fibre

The computer program spip is aimed at solving the Generalized Non-Linear Schrödinger equation (GNLSE), involved in optics e.g. in the modelling of light-wave propagation in an optical fibre, by the Interaction Picture method, a new efficient alternative method to the Symmetric Split-Step method. In the spip program a dedicated costless adaptive step-size control based on the use of a 4th order embedded Runge–Kutta method is implemented in order to speed up the resolution. Program summaryProgram title: SPIPCatalogue identifier: AEYQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Free software CeCILL license (see www.cecill.info/index.en.html)No. of lines in distributed program, including test data, etc.: 81174No. of bytes in distributed program, including test data, etc.: 6296651Distribution format: tar.gzProgramming language: C.Computer: Desktop computer.Operating system: Linux, MS Windows.RAM: 8 Giga bytesClassification: 4.12, 18.External routines:FFTW, a C subroutine library for computing the discrete Fourier transform, see [1] and http://www.fftw.org. Gnuplot, a portable command-line driven graphing utility, see [2] and http://www.gnuplot.info.Nature of problem: The program solves the Generalized Non-Linear Schrödinger Equation (GNLSE) which occurs in the field of non-linear optics as a model of wave propagation in fibre optics.Solution method: The GNLSE is solved by the Interaction Picture method coupled with an embedded Runge–Kutta scheme of order 4. The program includes a costless adaptive step-size control strategy taking advantage of the features of an embedded Runge–Kutta scheme designed for delivering a local error estimate at no extra-cost compared to the standard 4th order Runge–Kutta scheme.Running time: Highly dependent on the fibre length and accuracy required for the results. Typically between half a minute and several dozens of minutes.

Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

A space-time adaptive scheme is presented for solving advection equations in two space dimensions. The gradient-augmented level set method using a semi-Lagrangian formulation with backward time integration is coupled with a point value multiresolution analysis using Hermite interpolation. Thus locally refined dyadic spatial grids are introduced which are efficiently implemented with dynamic quadtree data structures. For adaptive time integration, an embedded Runge-Kutta method is employed. The precision of the new fully adaptive method is analysed and speed up of CPU time and memory compression with respect to the uniform grid discretization are reported.

Generalized Chebyshev Collocation Method

In this paper, we introduce a generalized Chebyshev collocation method (GCCM) based on the generalized Chebyshev polynomials for solving stiff systems. For employing a technique of the embedded Runge-Kutta method used in explicit schemes, the property of the generalized Chebyshev polynomials is used, in which the nodes for the higher degree polynomial are overlapped with those for the lower degree polynomial. The constructed algorithm controls both the error and the time step size simultaneously and further the errors at each integration step are embedded in the algorithm itself, which provides the efficiency of the computational cost. For the assessment of the effectiveness, numerical results obtained by the proposed method and the Radau IIA are presented and compared. Keywords—Generalized Chebyshev Collocation method,

GIP integrators for Matrix Riccati Differential Equations

Matrix Riccati Differential Equations (MRDEs) are initial value problems of the form:X′=A21-XA11+A22X-XA12X,X(0)=X0.These equations arise frequently throughout applied mathematics, science, and engineering. It can happen that even when the Aij are smooth functions of t or constant, the solution X may have a singularity or even infinitely many singularities.This paper shows several classes of numerical algorithms, which we call GIP integrators, that can solve for X past its singularities. Furthermore, none of the algorithms require knowledge of the placement or even existence of singularities in X. Also, it is shown how embedded Runge–Kutta methods can be used to construct GIP integrators to not only approximate X past singularities but also provide for error estimation to allow efficient time stepping. Finally, several examples are shown to validate the theory.

Embedded exponential operator splitting methods for the time integration of nonlinear evolution equations

In this work, we introduce embedded exponential operator splitting methods for the adaptive time integration of nonlinear evolution equations. In the spirit of embedded Runge–Kutta methods, such pairs of related higher-order split-step methods provide estimates of the local error with moderate additional computational effort as substeps of the basic integrator are reused to obtain a local error estimator.As illustrations, we construct a split-step pair of orders 4(3) involving real method coefficients, tailored for nonlinear Schrödinger equations, and two order 4(3) split-step pairs with complex method coefficients, appropriate for nonlinear parabolic problems. Our theoretical investigations and numerical examples show that the splitting methods retain their orders of convergence when applied to evolution problems with sufficiently regular solutions. Furthermore, we demonstrate the ability of the new algorithms to serve as a reliable basis for error control in the time integration of nonlinear evolution equations by applying them to the solution of two model problems, the two-dimensional cubic Schrödinger equation with focusing singularity and a three-dimensional reaction–diffusion equation. Moreover, we demonstrate the advantages of our real embedded 4(3) pair of splitting methods over a pair of unrelated schemes for the time-dependent Gross–Pitaevskii equation.

Cheap global error estimation in some Runge-Kutta pairs

The global error estimation and control problem is studied in embedded Runge–Kutta methods applied to ordinary differential equations (ODEs). We consider both nonstiff equations and stiff ones. Such problems arise in many areas of application and their accurate numerical solution is an important issue of applied mathematics. Usually, global error estimation and control techniques implemented in ODE solvers considerably extend execution time and they are rarely used in practical computations, especially, in implicit methods applied to stiff ODEs. In this paper we show that useful information can be obtained via simple summation of local error estimates calculated cheaply in the course of integration by some embedded Runge–Kutta pairs. Thus, since the embedded method local error estimation is a standard option of any practical solver for differential equations, our technique allows the global error estimation to be implemented naturally and without extra computations. Numerical tests presented here confirm this conclusion in practice. A comparison with explicit and implicit ODE solvers implemented with only local error control in MATLAB is also conducted.

RK-Slim: embedded Runge-Kutta without the excess baggage

Computationally demanding scientific simulations require numerical integration of large systems of ordinary differential equations (ODEs) in real time. Runge–Kutta (RK) methods with step-size control are frequently employed because they are accurate and have short execution times. Computational step sizes can be increased while maintaining accuracy if the error generated during each time step is below a predetermined threshold. However, if the error exceeds the threshold, the time step must be re-computed using a smaller step-size. In practice, the re-computation is unnecessary for most equations in the system because violations of the error tolerance are localized and occur in just a few equations. We present an efficient and accurate method for solving ODEs that exploits these observations on step-size by eliminating many unnecessary computations in embedded RK methods. We demonstrate how our new method can be a valuable tool for practitioners in the field through simulations with real-world data. Copyright © 2011 John Wiley & Sons, Ltd.

Two-point difference schemes of an arbitrary given order of accuracy for nonlinear BVPs

In this paper we consider difference schemes for two-point BVPs for systems of first order nonlinear ODEs. It was shown in former papers of the authors that starting from the two-point exact difference scheme (EDS) one can derive a so-called truncated difference scheme (TDS) which a priori possesses an arbitrary given order of accuracy m . Here, we demonstrate that the TDS can be reduced to the numerical solution of some IVPs defined on each segment [ x j − 1 , x j ] of the grid by an arbitrary IVP-solver of the order m . Using the difference schemes of the orders of accuracy m and m + 1 we develop an a posteriori error estimator for the numerical solution of the order m . An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. It is based on embedded Runge–Kutta methods. Some numerical results confirming the efficiency of the algorithm are given.

New embedded pairs of explicit Runge–Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems

A higher order Runge–Kutta (pair) method specially adapted to the numerical integration of IVPs with oscillatory solutions is presented. This method is based on the adapted methods proposed by Franco (see Ref. [J.M. Franco, Appl. Numer. Math. 50 (2004) 427]). We give explicit method (up to order 5) as well as pairs of embedded Runge–Kutta methods of order 5 and 4 designed using the FSAL properties. The stability of the new methods is analyzed. The numerical experiments are carried out to show the efficiency and robustness of our methods in comparison with some efficient methods.

A step size control algorithm for the weak approximation of stochastic differential equations

A vriable step size control algorithm for the weak approximation of stochastic differential equations is introduced. The algorithm is based on embedded Runge–Kutta methods which yield two approximations of different orders with a negligible additional computational effort. The difference of these two approximations is used as an estimator for the local error of the less precise approximation. Some numerical results are presented to illustrate the effectiveness of the introduced step size control method.

Embedded Runge–Kutta methods for the integration of a current control loop in an SRM dynamic finite element model

The use of embedded Runge–Kutta methods for the time-domain simulation of a dynamic finite element model of a switched reluctance motor is discussed, along with the features of embedded Runge–Kutta methods for the numerical solution of this model. It is shown that step size control and interpolator polynomials can be used to take accurately into account the discrete effects of current control and a pulse width modulated supply on the drive dynamics.