Applications of nonstandard finite difference schemes - download pdf or read online

By Ronald E Mickens

ISBN-10: 981024133X

ISBN-13: 9789810241339

This quantity could be divided into elements: a in basic terms mathematical half with contributions on finance arithmetic, interactions among geometry and physics and diversified components of arithmetic; one other half at the popularization of arithmetic and the location of girls in arithmetic Nonstandard finite distinction schemes / Ronald E. Mickens -- Nonstandard tools for advection-diffusion response equations / Hristo V. Kojouharov and Benito M. Chen -- program of nonstandard finite variations to unravel the wave equation and Maxwell's equations / James B. Cole -- Non-standard discretization equipment for a few organic types / H. Al-Kahby, F. Dannan, and S. Elaydi -- An advent to numerical integrators conserving actual homes / Martin J. Gander and Rita Meyer-Spasche

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Extra resources for Applications of nonstandard finite difference schemes

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110) To begin, note t h a t Eq. 109) has the following three sub-equations [48] ut+ux = 0, ux = buxx, ut = buxx. 113) A finite difference scheme t h a t incorporates the features of both of the sub-equations for Eq. 109) is M-i _ "m At uk u m uk ,, - 2uk + uk t C - u :m - l = b + Ai ft(eAx/6 _ 1 ) A a . 114) 42 Nonstandard Finite Difference Schemes with Arc = At. 117) or e Ax/6 At < _ j e Ai/6 + j Ax. 118) Note that when 6 = 0, this relation reduces to At < Ax. However, for this case, Eq. 114) reduces to Eq.

Since many of the mechanisms that give rise to numerical instabilities have their genesis in using a step-size that is larger than some relevant physical time scale, the above method of choosing eliminates these instabilities [24]. Stated another way, the use of such a denominator function allows the step-size h to be much larger than one normally used because as far as the discrete equations are concerned it is the effective step-size that determines the stability and not the ac­ tual step-size h.

Proceeding this way generally gives difference equations that do not have a first-integral. , have a first-integral, it is desirable to have this feature built into the discrete scheme, otherwise numerical instabilities will arise. One way to do this is to apply the nonstandard modeling rules directly to the energy function. 45) Nonstandard Finite Difference Schemes 32 where h = At, tk — hk, and <}>{h) has the property 0(h2). , E{xk,Xk-\) = E(xk-i,xk). 47) This constraint requires the nonlinear terms in the differential equa­ tion to be replaced by nonlocal representations in the finite differ­ ence model.

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Applications of nonstandard finite difference schemes by Ronald E Mickens

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