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One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10
The HBT(10)9 method for ODEs is expanded into HBT(10)9DAE for solving nonstiff and moderately stiff systems of fully implicit differential algebraic...
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Scale and modify for the second and third order BDF methods
This paper investigates a ‘scale and modify’ technique used with variable stepsize BDF methods. When the stepsize is changed using the usual scaling...
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G -matrices for algebraically stable general linear methodsThis paper describes a technique from Control whereby the G -matrix for an algebraically stable general linear method may be found in terms of the...
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Local theory of extrapolation methods
In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit...
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Numerically optimal Runge–Kutta pairs with interpolants
Explicit Runge–Kutta pairs are known to provide efficient solutions to initial value differential equations with inexpensive derivative evaluations....
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The BS class of Hermite spline quasi-interpolants on nonuniform knot distributions
The BS Hermite spline quasi-interpolation scheme is presented. It is related to the continuous extension of the BS linear multistep methods, a class...
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Stability of implicit-explicit linear multistep methods for ordinary and delay differential equations
Stability properties of implicit-explicit (IMEX) linear multistep methods for ordinary and delay differential equations are analyzed on the basis of...
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A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method
The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an...
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Multirate infinitesimal step methods for atmospheric flow simulation
The numerical solution of the Euler equations requires the treatment of processes in different temporal scales. Sound waves propagate fast compared...
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The existence of symplectic general linear methods
We derive a criterion that any general linear method must satisfy if it is symplectic. It is shown, by considering the method over several steps,...
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Algebraically stable general linear methods and the G-matrix
The standard algebraic stability condition for general linear methods (GLMs) is considered in a modified form, and connected to a branch of Control...
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Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations
In this paper, we focus on the error behavior of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations (NVDIDEs)...
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Runge–Kutta methods and viscous wave equations
We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an...
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One-step 9-stage Hermite–Birkhoff–Taylor ODE solver of order 10
A one-step 9-stage Hermite–Birkhoff–Taylor method of order 10, denoted by HBT(10)9, is constructed for solving nonstiff systems of first-order...
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Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods
The approximate preservation of quadratic first integrals (QFIs) of differential systems in the numerical integration with Runge–Kutta (RK) methods...
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Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise
The paper consists of two parts. In the first part of the paper, we proposed a procedure to estimate local errors of low order methods applied to...
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The Computational Complexity of Extrapolation Methods
This paper analyzes the cost of extrapolation methods for non-stiff ordinary differential equations depending on the number of digits of accuracy...
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Achieving Brouwer’s law with implicit Runge–Kutta methods
In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence...
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Local structure-preserving algorithms for partial differential equations
In this paper, we discuss the concept of local structure-preserving algorithms (SPAs) for partial differential equations, which are the natural...
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Exponential fitted Gauss, Radau and Lobatto methods of low order
Several exponential fitting Runge-Kutta methods of collocation type are derived as a generalization of the Gauss, Radau and Lobatto traditional...