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An Analysis of Rosenbrock Methods for Nonlinear Stiff Initial Value Problems
SIAM Journal on Numerical Analysis, 1982The paper presents an analysis of the Rosenbrock integration method applied to a stiff system of the form \[ (1)\qquad \dot x = f(t,x,y,\varepsilon ) + \varepsilon ^{ - 1} A(t)y,\quad \dot y = g(t,x,y,\varepsilon ) + \varepsilon ^{ - 1} \mu (t)By. \] This equation possesses the following desirable model properties.
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Stiff ODE Initial Value Problems
2011This chapter deals with stiff initial value problems for ODEs $$\dot{x}=F(x), \,x(0)={x_0}$$ The discretization of such problems is known to involve the solution of nonlinear systems per each discretization step–in one way or the other.
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Sensitivity Analysis of Stiff and Non-Stiff Initial-Value Problems
1998The solution y(t, t 0, y 0) of an initial-value problem (IVP) y(t) = f(t, y, p) with initial value y(t 0) = y 0 at a point t is a differentiable function of the initial value y 0 and the parameter vector p, provided f y and f p are continuous. The computation of the derivatives of y(t, t 0, y 0) plays an important role in the efficient numerical ...
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Numerical integrators for Stiff and Stiff oscillatory First Order initial value problems
Journal of the Nigerian Association of Mathematical Physics, 2008In this paper, efforts are geared towards the numerical solution of the first order initial value problem (I.V.P) of the form Y\' = F(X,Y), X∈[ a, b] , Y(a) = Y0, where Y\' is the total derivative of Y with respect to X.. The scheme so developed for the stated equation is in the same line of thought as Fatunla (1980). It is of order 6, L-stable
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Optimal extended one-step schemes of exponential type for stiff initial-value problems
International Journal of Computer Mathematics, 2004Extended one-step schemes of exponential type are introduced for the numerical solution of stiff initial-value problems. These schemes are uniformly convergent of third and fourth orders of accuracy. In addition, we show that these schemes are optimal when ϵ → 0. Numerical results and comparisons with other schemes are presented.
A. A. Salama, S. A. Bakr
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On the control of the global error in stiff initial value problems
1982An approach of B. Lindberg [3] to the estimation and control of a norm of the global error is modified and applied to systems of stiff ODE's in partitioned form.
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On the Convergence of Backward Differentiation Formulas for Stiff Initial Value Problems
BIT Numerical Mathematics, 2001The influence of a time-dependent transformation to a numerical method is studied. Thus convergence results of backward differentiation formulas applied to the non-autonomous stiff system y′ = A(t)y + Φ(t) are given. The approach is based on a special decomposition of the companion matrix.
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Stiffness in numerical initial-value problems: A and L-stability of numerical methods
International Journal of Mathematical Education in Science and Technology, 2001The main aim of this note is to help students to gain an insight into the asymptotic stability concept by means of visual representation of the stability regions of different numerical methods. This facilitates understanding of the meaning of stability for constant step sizes and the related concept of stiffness in numerical initial-value problems ...
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Some methods for separating stiff components in initial value problems
1984When solving a stiff differential system by an implicit method, factorizing the Jacobian and solving the resulting linear equations often dominate the cost. We develop some methods related to a block Schur factorization of the Jacobian for separating the stiff components.
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Generalized Runge-Kutta Processes for Stiff Initial-value Problems†
IMA Journal of Applied Mathematics, 1975Ehle, B. L., Lawson, J. D.
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