Partial Differential Equations in Software Printer barcode pdf417 in Software Partial Differential Equations

20. Partial Differential Equations using barcode integration for none control to generate, create none image in none applications. datamatrix When the indepen none for none dent variable u is a vector, the von Neumann analysis is slightly more complicated. For example, we can consider equation (20.1.

3), rewritten as @ vs @ r D (20.1.20) @t s @x vr The Lax method for this equation is rjnC1 D.

nC1 sj 1 n v t n .rj C1 C rjn 1 / C .s 2 2 x j C1 1 n v t n n D .sj C1 C sj 1 / C .r 2 2 x j C1 n sj 1 /. (20.1.21) rjn 1 /.

The von Neumann stability analysis now proceeds by assuming that the eigenmode is of the following (vector) form: n 0 rj n i kj x r (20.1.22) e n D sj s0 Here the vector on the right-hand side is a constant (both in space and in time) eigenvector, and is a complex number, as before.

Substituting (20.1.22) into (20.

1.21) and dividing by the power n , gives the homogeneous vector equation 2 3 2 3 2 3 v t i sin k x 7 6r 0 7 607 6.cos k x/ x (20.

1.23) 4 v t 5 4 5D4 5 0 i s0 sin k x .cos k x/ x This admits a solution only if the determinant of the matrix on the left vanishes, a condition easily shown to yield the two roots , D cos k x i v t sin k x x (20.

1.24). The stability co none for none ndition is that both roots satisfy j j 1. This again turns out to be simply the Courant condition (20.1.

17).. 20.1.3 Other Varieties of Error Thus far we have none for none been concerned with amplitude error, because of its intimate connection with the stability or instability of a differencing scheme. Other varieties of error are relevant when we shift our concern to accuracy, rather than stability. Finite difference schemes for hyperbolic equations can exhibit dispersion, or phase errors.

For example, equation (20.1.16) can be rewritten as v t i k x Ci 1 sin k x (20.

1.25) De x An arbitrary initial wave packet is a superposition of modes with different k s. At each timestep the modes get multiplied by different phase factors (20.

1.25), depending on their value of k. If t D x=v, then the exact solution for each mode of a wave packet f .

x vt / is obtained if each mode gets multiplied by exp. i k x/. For this value of t , equation (20.

1.25) shows that the nite difference solution gives. 20.1 Flux-Conservative Initial Value Problems the exact analyt none none ic result. However, if v t = x is not exactly 1, the phase relations of the modes can become hopelessly garbled and the wave packet disperses. Note from (20.

1.25) that the dispersion becomes large as soon as the wavelength becomes comparable to the grid spacing x. A third type of error is one associated with nonlinear hyperbolic equations and is therefore sometimes called nonlinear instability.

For example, a piece of the Euler or Navier-Stokes equations for uid ow looks like @v D @t v @v C ::: @x (20.1.26).

The nonlinear te rm in v can cause a transfer of energy in Fourier space from long wavelengths to short wavelengths. This results in a wave pro le steepening until a vertical pro le or shock develops. Since the von Neumann analysis suggests that the stability can depend on k x, a scheme that was stable for shallow pro les can become unstable for steep pro les.

This kind of dif culty arises in a differencing scheme where the cascade in Fourier space is halted at the shortest wavelength representable on the grid, that is, at k 1= x. If energy simply accumulates in these modes, it eventually swamps the energy in the long wavelength modes of interest. Nonlinear instability and shock formation are thus somewhat controlled by numerical viscosity such as that discussed in connection with equation (20.

1.18) above. In some uid problems, however, shock formation is not merely an annoyance, but an actual physical behavior of the uid whose detailed study is a goal.

Then, numerical viscosity alone may not be adequate or suf ciently controllable. This is a complicated subject that we discuss further in the subsection on uid dynamics, below. For wave equations, propagation errors (amplitude or phase) are usually most worrisome.

For advective equations, on the other hand, transport errors are usually of greater concern. In the Lax scheme, equation (20.1.

15), a disturbance in the advected quantity u at mesh point j propagates to mesh points j C 1 and j 1 at the next timestep. In reality, however, if the velocity v is positive, then only mesh point j C 1 should be affected. The simplest way to model the transport properties better is to use upwind differencing (see Figure 20.

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