Amplitude-shape method: The quasistatic method revisited

Mika, J.; Andrzejewski, Krzysztof; Parumasur, Nabendra

doi:10.1080/00411459808205631

Cited by 3 publications

(3 citation statements)

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“…As a first numerical example, we consider the following reaction diffusion equation. This simple example is chosen to illustrate the advantage of using the amplitude-shape approximation (ASA) (9) in comparison to the separation of variables approximation (SVA) (1). The linear partial differential equation is given by…”

Section: Numerical Modellingmentioning

confidence: 99%

“…Usually, v(x, t) is a slow varying function of time and is called the shape function because of its dependence on x, and (t) is a fast varying function of time and is called the amplitude. The resulting method is known as the amplitude-shape method [1][2][3][4]. The separation (2) was first used in the quasistatic method that was developed for nuclear engineering applications [5] and the amplitude-shape method may be viewed as an extension of this method.…”

Section: Introductionmentioning

confidence: 99%

“…Relative errors using(1) corresponding to the separation of variables approximation (SVA) and(9) corresponding to the amplitude-shape approximation (ASA).Copyright q 2007 John Wiley & Sons, Ltd. Math.…”

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confidence: 99%

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Amplitude–shape approximation as an extension of separation of variables

Parumasur

Singh

2007

Math Methods in App Sciences

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SUMMARYSeparation of variables is a well-known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude-shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well-established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi-implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude-shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function.

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Section: Numerical Modellingmentioning

confidence: 99%