Difference Equations from Differential Equations by Prof. Wilbert James Lick (auth.)

By Prof. Wilbert James Lick (auth.)

In computational mechanics, the 1st and ordinarily the main tough a part of an issue is the right kind formula of the matter. this can be often performed by way of differential equations. as soon as this formula is comprehensive, the interpretation of the governing differential equations into exact, strong, and bodily practical distinction equations could be a ambitious job. by means of comparability, the numerical assessment of those distinction equations on the way to receive an answer is mostly a lot less complicated. the current notes are essentially excited about the second one job, that of deriving actual, reliable, and bodily life like distinction equations from the governing differential equations. methods for the numerical assessment of those distinction equations also are provided. In later functions, the actual formula of the matter and the houses of the numerical answer, specifically as they're with regards to the numerical approximations inherent within the answer, are mentioned. there are lots of how one can shape distinction equations from differential equations.

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23 For fast, low viscosity flow, u» 1. In this case, the solution can be approximated by - e Ux = _ eU = e where z -u(l-x) =e -uz = 1 - x and is the distance from the point x = 1. 6) The solution decays to e- 1 in a distance given by uz= I or z = l/u. 7) The distance llu is usually called the boundary layer thickness. In order to derive difference equations corresponding to Eq. 1) by means of the integral method, first integrate Eq. 1) over the element, Xi-l/2:5. X:5. 8) This equation has a simple physical interpretation.

Its solution by the present integral method is discussed here. 30) u(oo) = I --- -_ -- ........... __ t:J - _ /J ..... 4 >~ 0 0 Q'l. 5-4. Solutions for ~ and y from Eqs. 20. 02 46 The general character of the solution to the above problem is as follows. Near the body at r = rIo the nonlinear term is negligible and the flow is dominated by the first two terms of Eq. 29), the viscous terms. 31) and the general solution is u = a + b In r. 33) A numerical solution to the above problem can readily be found by the usual finite difference procedure.

X:5. 8) This equation has a simple physical interpretation. The first two terms are the diffusive fluxes of across the two boundaries at xi+l/2 and xi-l/2 while the last two terms are the convective fluxes across these same boundaries. The sum of the fluxes is zero in this case since there are no internal sources of <1>. To proceed further and obtain a difference equation, the functions appearing in Eq. 8) must be approximated. If this is done by assuming is linear in each sub-interval, Xi :5.

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