# Addition theorems; the addition theorems of group theory and by Henry B. Mann.

By Henry B. Mann.

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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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