A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and parallel
نویسندگان
چکیده
Citation A comparison of high-order explicit Runge–Kutta, extrapolation, and deferred correction methods in serial and We compare the three main types of high-order one-step initial value solvers: extrapolation, spectral deferred correction, and embedded Runge–Kutta pairs. We consider orders four through twelve, including both serial and parallel implementations. We cast extrapolation and deferred correction methods as fixed-order Runge–Kutta methods, providing a natural framework for the comparison. The stability and accuracy properties of the methods are analyzed by theoretical measures, and these are compared with the results of numerical tests. In serial, the eighth-order pair of Prince and Dormand (DOP8) is most efficient. But other high-order methods can be more efficient than DOP8 when implemented in parallel. This is demonstrated by comparing a parallelized version of the well-known ODEX code with the (serial) DOP853 code. For an N-body problem with N = 400, the experimental extrapolation code is as fast as the tuned Runge–Kutta pair at loose tolerances, and is up to two times as fast at tight tolerances.
منابع مشابه
Parallel High-Order Integrators
In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multi-core architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on Integral Deferred Correction (IDC), which was itself motivated by Spectral Deferred Correction by Dutt, Greengard and Rokhlin (BIT-2000). The ...
متن کاملStepwise Global Error Control in an Explicit Runge-Kutta Method Using Local Extrapolation with High-Order Selective Quenching
Received: December 9, 2010 Accepted: January 6, 2011 doi:10.5539/jmr.v3n2p126 Abstract Stepwise local error control using local extrapolation in Runge-Kutta methods is well-known. In this paper, we introduce an algorithm, designated RKrvQz, that is capable of controlling local and global errors in a stepwise manner. The algorithm utilizes three Runge-Kutta methods, of orders r, v and z, with r ...
متن کاملIntegral deferred correction methods constructed with high order Runge-Kutta integrators
Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin [5]. It was shown in [5] that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff a...
متن کاملPreliminary Experiences with Extrapolation Methods for Parallel Solution of Differential Algebraic Equations
This paper discusses the use of extrapolation methods for the parallel solution of diierential algebraic equations. The DAEs investigated are implicit and have explicit constrains and the underlying methods used for the extrapolation are Runge-Kutta methods. An implementation is described and preliminary results are presented.
متن کاملComments on High Order Integrators Embedded within Integral Deferred Correction Methods
Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin, [3]. In this paper, we study the properties of these integral deferred correction methods, constructed using high order integrators in the prediction and correction loops, and various distributions of quadrature nodes. The smoothness of the error vector a...
متن کامل