scholarly journals Strong Stability Preserving IMEX Methods for Partitioned Systems of Differential Equations

2021 ◽  
Author(s):  
Giuseppe Izzo ◽  
Zdzisław Jackiewicz
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Order Of Convergence ◽  
Large Range ◽  
Strong Stability ◽  
Strong Stability Preserving ◽  
Systems Of Differential Equations ◽  
Imex Methods ◽  
Stage Order

AbstractWe investigate strong stability preserving (SSP) implicit-explicit (IMEX) methods for partitioned systems of differential equations with stiff and nonstiff subsystems. Conditions for order p and stage order $$q=p$$ q = p are derived, and characterization of SSP IMEX methods is provided following the recent work by Spijker. Stability properties of these methods with respect to the decoupled linear system with a complex parameter, and a coupled linear system with real parameters are also investigated. Examples of methods up to the order $$p=4$$ p = 4 and stage order $$q=p$$ q = p are provided. Numerical examples on six partitioned test systems confirm that the derived methods achieve the expected order of convergence for large range of stepsizes of integration, and they are also suitable for preserving the accuracy in the stiff limit or preserving the positivity of the numerical solution for large stepsizes.

scholarly journals STRONG STABILITY PRESERVING MULTISTAGE INTEGRATION METHODS

2015 ◽  
Vol 20 (5) ◽  
pp. 552-577 ◽  
Author(s):  
Giuseppe Izzo ◽  
Zdzislaw Jackiewicz
Keyword(s):  
Strong Stability ◽  
General Linear Methods ◽  
General Linear ◽  
Strong Stability Preserving ◽  
Order Of Accuracy ◽  
Integration Methods ◽  
Numerical Tests ◽  
Stage Order

In this paper we systematically investigate explicit strong stability preserving (SSP) multistage integration methods, a subclass of general linear methods (GLMs), of order p and stage order q ≤ p. Characterization of this class of SSP GLMs is given and examples of SSP methods of order p ≤ 4 and stage order q = 1, 2, . . . , p are provided. Numerical tests are reported which confirm that the constructed methods achieve the expected order of accuracy and preserve monotonicity.

scholarly journals Bilateral approximations and periodic solutions of systems of differential equations with impulses

1985 ◽  
Vol 31 (2) ◽  
pp. 293-307
Author(s):  
S.G. Hristova ◽  
D.D. Bainov
Keyword(s):  
Periodic Solution ◽  
Linear System ◽  
Differential Equations ◽  
Periodic Solutions ◽  
System Of Differential Equations ◽  
Systems Of Differential Equations ◽  
Non Linear ◽  
Impulsive Perturbations ◽  
Non Linear System

The paper justifies a method of bilateral approximations for finding the periodic solution of a non-linear system of differential equations with impulsive perturbations at fixed moments of time.

scholarly journals Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Basile Morcrette ◽  
Hosam M. Mahmoud
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Probability Generating Function ◽  
Isomorphism Theorem ◽  
Systems Of Differential Equations ◽  
Exactly Solvable ◽  
Exact Distributions ◽  
International Audience ◽  
Weighted Probability

International audience This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.

scholarly journals Rational functions with maximal radius of absolute monotonicity

2014 ◽  
Vol 17 (1) ◽  
pp. 159-205 ◽  
Author(s):  
Lajos Lóczi ◽  
David I. Ketcheson
Keyword(s):  
Initial Value Problems ◽  
Rational Functions ◽  
Strong Stability ◽  
Strong Stability Preserving ◽  
Algebraic Numbers ◽  
Initial Value ◽  
Stage Order ◽  
Absolute Monotonicity ◽  
Runge Kutta ◽  
Radius Of Absolute Monotonicity

AbstractWe study the radius of absolute monotonicity $R$ of rational functions with numerator and denominator of degree $s$ that approximate the exponential function to order $p$. Such functions arise in the application of implicit $s$-stage, order $p$ Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$ and $R>2s$, disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.

scholarly journals Construction of an autogenerator dynamic model applicable to nuclear processes

2011 ◽  
Vol 26 (1) ◽  
pp. 74-77
Author(s):  
Diana Dolicanin ◽  
Vladimir Amelkin ◽  
Milisav Stefanovic ◽  
Milos Vujisic
Keyword(s):  
Mathematical Model ◽  
Vector Field ◽  
Linear System ◽  
Differential Equations ◽  
Dynamic Model ◽  
New Method ◽  
Oscillation Regime ◽  
Systems Of Differential Equations ◽  
Non Linear System ◽  
Nuclear Processes

We propose a new method for constructing a mathematical model of a non-linear system in an auto-oscillation regime. The method is based on the divergence of a vector field having a constant value along the corresponding periodical motion. The variants of the obtained model could be used for describing nuclear processes that are represented by the systems of differential equations analogous to that of the presented model.

ACCELERATION OF CONVERGENCE OF WAVEFORM RELAXATION ITERATIONS FOR SECOND ORDER LINEAR SYSTEM

1996 ◽  
Vol 20 (1) ◽  
pp. 87-103
Author(s):  
Z. Jackiewicz ◽  
J. Knap
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Linear Systems ◽  
Second Order ◽  
Waveform Relaxation ◽  
Relaxation Techniques ◽  
Speed Of Convergence ◽  
Acceleration Of Convergence ◽  
Order Linear ◽  
Systems Of Differential Equations

The solution of second order linear systems of differential equations by waveform relaxation techniques is investigated. It is demonstrated that overlapping of components of the system improves significantly the speed of convergence of the resulting waveform relaxation iterations.

SOLUTION OF SOME SYSTEMS OF DIFFERENTIAL EQUATIONS IN MATHEMATICAL SYSTEM MAPLE

2013 ◽  
Vol 1 (05) ◽  
pp. 58-65
Author(s):  
Yunona Rinatovna Krakhmaleva ◽  
◽  
Gulzhan Kadyrkhanovna Dzhanabayeva ◽  
Keyword(s):  
Differential Equations ◽  
Systems Of Differential Equations ◽  
Mathematical System

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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