scholarly journals Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions

2009 ◽  
Vol 46 (2) ◽  
pp. 604-620 ◽  
Author(s):  
G. A. Panopoulos ◽  
Z. A. Anastassi ◽  
T. E. Simos
Keyword(s):  
Initial Value Problems ◽  
Implicit Methods ◽  
Initial Value ◽  
Oscillating Solutions

EXPLICIT EIGHTH ORDER NUMEROV-TYPE METHODS WITH REDUCED NUMBER OF STAGES FOR OSCILLATORY IVPs

2008 ◽  
Vol 19 (06) ◽  
pp. 957-970 ◽  
Author(s):  
I. Th. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
Computational Cost ◽  
Phase Lag ◽  
Initial Value ◽  
Eighth Order ◽  
Oscillatory Solutions ◽  
Numerical Tests ◽  
New Methods ◽  
Oscillating Solutions ◽  
Algebraic Order

Using a new methodology for deriving hybrid Numerov-type schemes, we present new explicit methods for the solution of second order initial value problems with oscillating solutions. The new methods attain algebraic order eight at a cost of eight function evaluations per step which is the most economical in computational cost that can be found in the literature. The methods have high amplification and phase-lag order characteristics in order to suit to the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

scholarly journals Multirate linearly-implicit GARK schemes

2021 ◽  
Author(s):  
Michael Günther ◽  
Adrian Sandu
Keyword(s):  
Numerical Stability ◽  
Initial Value Problems ◽  
Time Integration ◽  
Order Conditions ◽  
Systems Of Equations ◽  
Implicit Methods ◽  
Initial Value ◽  
Multirate Time Integration ◽  
Efficient Coupling ◽  
Linear Systems Of Equations

AbstractMany complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

scholarly journals Implicit Methods for Numerical Solution of Singular Initial Value Problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Md. Rahaman Habibur ◽  
M. Kamrul Hasan ◽  
Md. Ayub Ali ◽  
M. Shamsul Alam
Keyword(s):  
Singular Point ◽  
Numerical Solution ◽  
Initial Value Problems ◽  
Accurate Result ◽  
Implicit Method ◽  
Implicit Methods ◽  
Initial Value ◽  
Runge Kutta

AbstractVarious order of implicit method has been formulated for solving initial value problems having an initial singular point. The method provides better result than those obtained by used implicit formulae developed based on Euler and Runge-Kutta methods. Romberg scheme has been used for obtaining more accurate result.

EXPONENTIALLY AND TRIGONOMETRICALLY FITTED EXPLICIT ADVANCED STEP-POINT (EAS) METHODS FOR INITIAL VALUE PROBLEMS WITH OSCILLATING SOLUTIONS

2003 ◽  
Vol 14 (02) ◽  
pp. 175-184 ◽  
Author(s):  
G. PSIHOYIOS ◽  
T. E. SIMOS
Keyword(s):  
Numerical Solution ◽  
Initial Value Problems ◽  
Initial Value ◽  
Numerical Examples ◽  
Oscillating Solutions ◽  
Exponentially Fitted ◽  
Predictor Corrector ◽  
Area Of Application

In this paper, an exponentially fitted and trigonometrically fitted predictor–corrector class of methods is developed. These methods represent a totally new area of application for the explicit advanced step-point or EAS methods developed by Psihoyios and Cash. Numerical examples show that the newly developed procedure is much more efficient than well-known methods for the numerical solution of initial value problems with oscillating solutions.

NUMEROV-TYPE METHODS FOR OSCILLATORY LINEAR INITIAL VALUE PROBLEMS

2009 ◽  
Vol 20 (03) ◽  
pp. 383-398 ◽  
Author(s):  
I. TH. FAMELIS
Keyword(s):  
Initial Value Problems ◽  
New Method ◽  
Phase Lag ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
Type Method ◽  
Numerical Tests ◽  
Oscillating Solutions ◽  
Algebraic Order ◽  
High Phase

We present a new explicit Numerov-type method for the solution of second-order linear initial value problems with oscillating solutions. The new method attains algebraic order seven at a cost of six function evaluations per step. The method has the characteristic of zero dissipation and high phase-lag order making it suitable for the solution of problems with oscillatory solutions. The numerical tests in a variety of problems justify our effort.

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