Differential finite magnitude

We note that bdb is related to the classical differential scattering cross section. It is a short-hand notation for b(x) di>/d/ dx the magnitude of the derivative di>/dx falls off to zero quickly so that the integral over bdb is actually finite. 

Such a partial differential equation, expressed in terms of discrete units of x and t, is called a finite difference equation. It is subject to some discretization error depending on the magnitude of the finite differences chosen. In the limit as x and t become infinitesimal, the equation converges to reality (i.e., the discretization error becomes zero) and thus... 

The most important feature of this dispersion-optimized FDTD method is the higher order nonstandard finite-difference schemes [6, 7] that substitute their conventional counterparts in the differentiation of Ampere s and Faraday s laws, as already described in (3.31). The proposed technique can be occasionally even 7 to 8 orders of magnitude more accurate than the fourth-order implementations of Chapter 2. Although the cost is slightly increased, the overall simulation benefits from the low resolutions and the reduced number of iterations. Thus, for spatial derivative approximation, the following two operators are defined ... 

Since Eq. (4.17) becomes more nearly exact as Ax is made smaller, we make it into an exact equation by making Ax become smaller than any finite quantity that anyone can name. We do not make Ax strictly vanish, but we say that we make it become infinitesimal. That is, we make it smaller in magnitude than any nonzero quantity one might specify. In this limit. Ax is called the differential dx and we write ... 

The quantum theory replaces the classical differential coefficient by a difference quotient. We do not proceed to the limit of infinitely small variations of the independent variables, but stop at finite intervals of magnitude h. 

The existence of truncation errors in finite difference approximations to differential equations is discussed in numerical analysis texts with respect to round-off error and computational instabilities (Roache, 1972 Richtmyer and Morton, 1957), but Lantz (1971) was among the first to address the form of the truncation error as it related to diffusion. Lantz considered a linear, convective, parabolic equation similar to 9u/9t + U 9u/9x = e S u/Sx and differenced it in several ways. He showed that the effective diffusion coefficient was not 8, as one might have suggested analytically, but 8 + 0(Ax, At) (so that the actual diffusion term appearing in computed solutions is the modified coefficient times c2u/9x2) where the 0(Ax,At) truncation errors, being functions of u(x,t), are comparable in magnitude to 8. Because this artificial diffusion necessarily differs from the actual physical model, one would expect that the entropy conditions characteristic of the computed results could likely be fictitious. 

The underlying Finite Volume Method divides the system geometry into small (linked) partial volumes of the same magnitude analog to the well-known finite elements in FEA (Finite Element Analysis). The differential equations that describe the flow are converted in this way into difference equations (integrals to be summed). These can then be solved as a linear equation system using a high performance Solver. If this calculation is repeated for each time step, the result is a description of the time-dependent transient flow. 

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