9 research outputs found
An eighth-order exponentially fitted two-step hybrid method of explicit type for solving orbital and oscillatory problems
The construction of an eighth-order exponentially fitted (EF) two-step hybrid method for the numerical integration of oscillatory second-order initial value problems (IVPs) is considered. The EF two-step hybrid methods integrate exactly differential systems whose solutions can be expressed as linear combinations of exponential or trigonometric functions and have variable coefficients depending on the frequency of each problem. Based on the order conditions and the EF conditions for this class of methods, we construct an explicit EF two-step hybrid method with symmetric nodes and algebraic order eight which only uses seven function evaluations per step. This new method has the highest algebraic order we know for the case of explicit EF two-step hybrid methods. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order EF scheme is more efficient than other standard and EF two-step hybrid codes recently proposed in the scientific literature
High Order Multistep Methods with Improved Phase-Lag Characteristics for the Integration of the Schr\"odinger Equation
In this work we introduce a new family of twelve-step linear multistep
methods for the integration of the Schr\"odinger equation. The new methods are
constructed by adopting a new methodology which improves the phase lag
characteristics by vanishing both the phase lag function and its first
derivatives at a specific frequency. This results in decreasing the sensitivity
of the integration method on the estimated frequency of the problem. The
efficiency of the new family of methods is proved via error analysis and
numerical applications.Comment: 36 pages, 6 figure
A class of explicit high-order exponentially-fitted two-step methods for solving oscillatory IVPs
The derivation of new exponentially fitted (EF) modified two-step hybrid (MTSH) methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods are modifications of classical two-step hybrid methods so that they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(¿t), exp(-¿t)}, ¿¿C, or equivalently {sin(¿t), cos(¿t)} when ¿=i¿, ¿¿R, where ¿ represents an approximation of the main frequency of the problem. The EF conditions and the conditions for this class of EF schemes to have algebraic order p (with p=8) are derived. With the help of these conditions we construct explicit EFMTSH methods with algebraic orders seven and eight which require five and six function evaluation per step, respectively. These new EFMTSH schemes are optimal among the two-step hybrid methods in the sense that they reach a certain order of accuracy with minimal computational cost per step. In order to show the efficiency of the new high order explicit EFMTSH methods in comparison to other EF and standard two-step hybrid codes from the literature some numerical experiments with several orbital and oscillatory problems are presented
Exponentially fitted two-step hybrid methods for y''=f(x,y)
It is the purpose of this paper to derive two-step hybrid methods for y '' = f(x, y), with oscillatory or periodic solutions, specially tuned to the behaviour of the solution, through the usage of the exponential fitting technique. The construction of two-step exponentially fitted hybrid methods is shown and their properties are discussed. Some numerical experiments confirming the theoretical expectations are provided
A Trigonometrically Fitted Block Method for Solving Oscillatory Second-Order Initial Value Problems and Hamiltonian Systems
In this paper, we present a block hybrid trigonometrically fitted Runge-Kutta-Nyström method (BHTRKNM), whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including Hamiltonian systems such as the energy conserving equations and systems arising from the semidiscretization of partial differential equations (PDEs). Four discrete hybrid formulas used to formulate the BHTRKNM are provided by a continuous one-step hybrid trigonometrically fitted method with an off-grid point. We implement BHTRKNM in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHTRKNM is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages
Solving the Telegraph and Oscillatory Differential Equations by a Block Hybrid Trigonometrically Fitted Algorithm
We propose a block hybrid trigonometrically fitted (BHT) method, whose coefficients are functions of the frequency and the step-size for directly solving general second-order initial value problems (IVPs), including systems arising from the semidiscretization of hyperbolic Partial Differential Equations (PDEs), such as the Telegraph equation. The BHT is formulated from eight discrete hybrid formulas which are provided by a continuous two-step hybrid trigonometrically fitted method with two off-grid points. The BHT is implemented in a block-by-block fashion; in this way, the method does not suffer from the disadvantages of requiring starting values and predictors which are inherent in predictor-corrector methods. The stability property of the BHT is discussed and the performance of the method is demonstrated on some numerical examples to show accuracy and efficiency advantages
Numerical treatment of special second order ordinary differential equations: general and exponentially fitted methods
2010 - 2011The aim of this research is the construction and the analysis of new families of numerical
methods for the integration of special second order Ordinary Differential Equations
(ODEs). The modeling of continuous time dynamical systems using second order ODEs
is widely used in many elds of applications, as celestial mechanics, seismology, molecular
dynamics, or in the semidiscretisation of partial differential equations (which leads to
high dimensional systems and stiffness). Although the numerical treatment of this problem
has been widely discussed in the literature, the interest in this area is still vivid,
because such equations generally exhibit typical problems (e.g. stiffness, metastability,
periodicity, high oscillations), which must efficiently be overcome by using suitable
numerical integrators. The purpose of this research is twofold: on the one hand to construct
a general family of numerical methods for special second order ODEs of the type
y00 = f(y(t)), in order to provide an unifying approach for the analysis of the properties
of consistency, zero-stability and convergence; on the other hand to derive special
purpose methods, that follow the oscillatory or periodic behaviour of the solution of the
problem...[edited by author]X n. s