1,721,038 research outputs found
General Linear Methods with Large Regions of Absolute Stability
No full textWe describe the construction of general linear methods in Nordsieck form of order p and stage order q = p with large regions of absolute stability. We review the concepts of Runge-Kutta and quadratic stability and inherent Runge-Kutta and inherent quadratic stability [3] which aid in
the construction of general linear methods with desirable stability properties. We also derive the representation formulas for some of the coefficient matrices of Nordsieck methods [1]. The search for these methods is based on maximizing the area of the intersection of the region of absolute stability with the negative complex plane by various optimization routines [2]. We first construct
quadratic polynomials with large regions of absolute stability and then search for methods whose stability functions matches these stability polynomials. The efficient computation of coefficients of these polynomials utilizes the fast Fourier transform. Examples of methods obtained in this way are presented up to the order six. This is a joint work with A. Cardone from University of Salerno, and H. Mittelmann from Arizona State University.
REFERENCES
[1] A. Cardone and Z. Jackiewicz. Explicit Nordsieck methods with quadratic stability. Submitted.
[2] A. Cardone, Z. Jackiewicz and H. Mittelmann. Search for Nordsieck methods of high order with quadratic stability.
Manuscript.
[3] Z. Jackiewicz. General Linear Methods for Ordinary Differential Equations. John Wiley & Sons, Hoboken, New
Jersey, 2009
Analysis and Practical Construction of Two-Step Runge-Kutta Methods for Ordinary Differential Equations
No full textIn recent years some attention has been devoted to two-step Runge-Kutta methods (TSRK), in order to achieve classes of methods useful for the numerical solution of real problems. The main topic of this talk is the practical development of TSRK methods for the numerical solution of Ordinary Differential Equations (ODEs). Inspired by recent works on General Linear Methods, we derive the class of TSRK methods with Inherent Quadratic Stability (IQS), i.e. TSRK methods with quadratic stability function, in order to show an algorithmic construction of TSRK methods with strong stability properties. In particular, we deal with the one point spectrum and continuous TSRK with IQS
Two-step almost collocation methods for Volterra integral equations
Full text linkIn this paper we construct a new class of continuous methods for Volterra integral equations.
These methods are obtained by using a collocation technique and by relaxing some of the collocation conditions in order to obtain good stability properties
Strong Stability Preserving General Linear Methods
No full textWe describe the construction of strong stability preserving (SSP) general linear methods (GLMs) for ordinary differential equations. This construction is based on the monotonicity criterion for SSP methods. This criterion can be formulated as a minimization problem, where the objective function depends on the Courant-Friedrichs-Levy (CFL) coefficient of the method, and the nonlinear constrains depend on the unknown remaining parameters of the methods. The solution to this constrained minimization problem leads to new SSP GLMs of high order and stage order
Efficient general linear methods for non-stiff differential equations
No full textThe aim of our research is the construction and analysis of efficient gene-
ral linear methods (GLM), which achieve a good balance between accuracy
and stability properties
The Class of Implicit-explicit General Linear Methods for Ordinary Differential Equations
No full textThe class of implicit-explicit (IMEX) methods are numerical scheme designed for numerical solution of ordinary differential systems with splitting of the right hand sides of the differential systems into two parts, one of which is non-stiff or mildly stiff, and suitable for explicit time integration, and the other part is stiff, and suitable for implicit time integration.
We described the construction of IMEX general linear methods with desired stability properties. Under suitable assumptions such as the form of coefficient matrices, order and stage order of methods, the form of stability function we attempt to maximize the combined region of absolute stability. Finally, we apply constructed methods to a series of test problems.
References
[1] M. Bras, A. Cardone, Z. Jackiewicz, P. Pierzchala, Error propagation for implicit-explicit general linear methods, Appl. Numer. Math., 131 (2018), pp. 207–231.
[2] A. Cardone, Z. Jackiewicz, A. Sandu, H. Zhang, Extrapolation-based implicit–explicit general linear methods, Numer. Algorithms, 65 (2014), pp. 377—399
Construction of strong stability preserving general linear methods
No full textWe describe a tool for investigating the strong stability preserving (SSP) general linear methods (GLMs) with two external stages and s internal stages, and derive example of methods which have larger effective Courant-Friedrichs-Levy coefficients than the class of two-step Runge-Kutta (TSRK) methods introduced by Jackiewicz and Tracogna, whose SSP properties were analyzes recently by Ketcheson, Gottlieb, and MacDonald. Numerical examples illustrate that the class of methods derived in this paper achieve the expected order of accuracy and do not produce spurious oscillations for discretizations of hyperbolic conservation laws, when combined with appropriate discretizations in spatial variables
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