Applications of nonstandard finite difference schemes by Ronald E Mickens

By Ronald E Mickens

This quantity should be divided into components: a only mathematical half with contributions on finance arithmetic, interactions among geometry and physics and diverse parts of arithmetic; one other half at the popularization of arithmetic and the placement of ladies in arithmetic Nonstandard finite distinction schemes / Ronald E. Mickens -- Nonstandard equipment for advection-diffusion response equations / Hristo V. Kojouharov and Benito M. Chen -- software of nonstandard finite alterations to resolve the wave equation and Maxwell's equations / James B. Cole -- Non-standard discretization tools for a few organic types / H. Al-Kahby, F. Dannan, and S. Elaydi -- An creation to numerical integrators keeping actual houses / Martin J. Gander and Rita Meyer-Spasche

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Eq. 80), can be gotten by applying the Applications 41 difference operator, A, to Eq. 85). 107) where ip(h) has the property 0{h2). 109) where the constant b is positive. This equation has both linear advection and diffusion. In general, the independent variable is such t h a t it must be non-negative. This positivity property will be used later t o obtain a functional relationship between the step-sizes At and Ax, where xm = (Ax)m, tk = (At)k, t 4 - «(xm,*i). 110) To begin, note t h a t Eq. 109) has the following three sub-equations [48] ut+ux = 0, ux = buxx, ut = buxx.

Numerical instabilities can arise because the discrete equa­ tions do not satisfy a principle or condition that is of critical importance for the corresponding solutions of the differential equations. An important example is the condition of positivity that must be satisfied by many sys­ tems in the natural and engineering sciences. If the discrete equations allow their solutions to become negative, then numerical instabilities will occur. The existence of a first-integral is another important condition for many systems.

Mickens, "Exact solutions to a finite difference model of a nonlinear reaction-advection equation: Implications for numerical analysis," Numer­ ical Methods for Partial Differential Equations 5 (1989), 313-325. [2] R. E. Mickens, Nonstandard Finite Difference Models of Differentials (World Scientific, Singapore, 1994). [3] F. B. Hildebrand, Finite-Difference Equations and Simulations (Prentice-Hall, Englewood Cliffs, NJ, 1968). [4] J. M. Ortega and W. G. , An Introduction to Numerical Methods for Differential Equations (Pitman, Marshfield, MA, 1981).

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