Showing papers on "L-stability published in 1997"

A delay differential equation solver based on a continuous Runge-Kutta method with defect control

Abstract: We have recently developed a generic approach for solving neutral delay differential equations based on the use of a continuous Runge–Kutta formula with defect control and investigated its convergence properties. In this paper, we describe a method, DDVERK, which implements this approach and justify the strategies and heuristics that have been adopted. In particular we show how the assumptions related to error control, stepsize control, and discontinuity detection (required for convergence) can be efficiently realized for a particular sixth-order numerical method. Summaries of extensive testing are also reported.

Simulation of ordinary differential equations on manifolds : some numerical experiments and verifications

Abstract: During the last few years, different approaches for integrating ordinary differential equations on manifolds have been published. In this work, we consider two of these approaches. We present some numerical experiments showing benefits and some pitfalls when using the new methods. To demonstrate how they work, we compare with well known classical methods, e.g. Newmark and Runge-Kutta methods.

Runge–Kutta characteristic methods for first‐order linear hyperbolic equations

Abstract: We develop two Runge–Kutta characteristic methods for the solution of the initial-boundary value problems for first-order linear hyperbolic equations. One of the methods is based on a backtracking of the characteristics, while the other is based on forward tracking. The derived schemes naturally incorporate inflow boundary conditions into their formulations and do not need any artificial outflow boundary condition. They are fully mass conservative and can be viewed as higher-order time integration schemes improved over the ELLAM (Eulerian–Lagrangian localized adjoint method) method developed previously. Moreover, they have regularly structured, well-conditioned, symmetric, and positive-definite coefficient matrices. Extensive numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the Petrov–Galerkin methods, the streamline diffusion methods, the continuous and discontinuous Galerkin methods, the MUSCL, and the ENO schemes. The numerical experiments also verify the optimal-order convergence rates of the Runge–Kutta methods developed in this article. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 617–661, 1997

Order and stability of parallel methods for stiff problems

Abstract: This paper explores methods of DIMSIM structure, with coefficient matrix of the form A = λ I, allowing the stages to be evaluated in parallel. It is found that many of these methods possess a strong A-stable property making them suitable for the solution of large stiff problems.

The Stability of Natural Runge-Kutta Methods for Nonlinear Delay Differential Equations

Abstract: A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any collocation method is equivalent to one of such methods. In this paper, stability properties of natural Runge-Kutta methods are studied using nonlinear DDEs which have a quadratic Liapunov functional. A discrete analogue of the functional is defined for each method, and the stability of the method is examined on the basis of this analogue. In particular, it is shown that an algebracially stable method, if it satisfies an additional condition, preserves the asymptotic properties of the original equations for every stepsize.

A variable-step variable-order algorithm for systems of stiff ODEs

Abstract: A variable-step variable-order algorithm for stiff ODEs based on previously derived stabilized extended one-step methods is established. The developed code is tested on certain initial-value problems for systems of ODEs contained in the test set proposed by CWI.

Using Diagonally Implicit Multistage Integration Methods for Solving Ordinary Differential Equations. Part 2: Implicit Methods.

Abstract: : New theoretical results for implicit methods of the recently invented and highly promising class of diagonally implicit multistage integration methods (DIMSIMs) are described. Some new A-stable and L-stable methods are derived, and conditions for a method to have an A-stable or L-stable FASAL (First Approximately Same As Last) implementation are derived. New alternative error estimates are proposed and successfully tested. The stability regions for alternative predictor-corrector DIMSIM implementations are derived. Implementation parameters for stiff solvers based on second and fifth order DIMSIMs are derived. Development of prototype stiff solvers of both second and fifth order stiff DIMSIM solvers is described. Results of successful tests on the Prothero-Robinson problem are reported and demonstrate that DIMSIMs may be used to develop efficient stiff solvers.

L-decremented ROW methods of the third order of accuracy

Abstract: A complete analysis of the three- and four-stage ROW methods is carried out. Families of schemes of the third order of accuracy are obtained with the first and second order of L-decrementation of stiff components. A new algorithm is proposed for estimating the local error and for the automatic step control. Examples of specific schemes are presented. These schemes are compared with the best (in the class of ROW methods) solver for numerical solution of stiff systems of ordinary differential equations, ROS4A. The advantage of the L-decremented methods is demonstrated for a complicated test problem characterized by high stiffness (the Van der Pol nonlinear oscillator).

A modified Taylor series method for solving initial-value problems in ordinary differential equations

Abstract: A one-step algorithm for solving ordinary differential equations with initial conditions is developed. Given the p first derivatives of y(x), the unknown function determined by the differential equation, this algorithm provides a method with a local discretization error of order 2p+1 instead of p+1 as is to be expected if we use the standard Taylor method with p derivatives. Several examples are given which illustrate the behaviour of the proposed algorithm and are also used to compare the efficiency of this algorithm, in terms of the computer time needed to attain a given final error tolerance, with other commonly applied techniques, such as Runge-Kutta or Adams-Moulton. The range of applicability of the method is also considered.

An efficient semi-implicit time accurate scheme for unsteady viscous flow on dynamic grids

Abstract: This paper presents a semi-implicit method for efficient and high-order time-accurate computations for unsteady viscous hypersonic flows over fixed or moving bodies using Navier-Stokes equations. The equations are discretized in space using a second-order TVD scheme on moving structured grids. If explicit schemes are used to advance the equations in time, the small grid sizes in the wall-normal direction in the boundary layers imposed severe restrictions on the time steps. In the current method, the spatial discretization of the governing equations is separated into stiff terms involving derivatives along the wall-normal direction and nonstiff terms for the rest of the equations. The split equations are then advanced in time using second- and third-order semi-implicit Runge-Kutta schemes so that the nonstiff streamwise terms are treated by explicit Runge-Kutta methods and stiff wall-normal terms are simultaneously treated by implicit Runge-Kutta methods. The semi-implicit method leads to block pentagonal diagonal systems of implicit equations that can be solved efficiently. (Author)