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Vice Chair, West Virginia University School of Medicine The exact solution is a mere translation from the initial position (dashed curve on the left) by 50 grid points downstream (dash-dotted curve on the right) treatment xanax withdrawal order agarol laxative 120ml. The numerical method generates a solution that is roughly similar to the exact solution symptoms toxic shock syndrome best 120ml agarol laxative, with the solution varying around the correct value cold medications trusted 120ml agarol laxative. Physically medications like prozac trusted agarol laxative 120ml, however, the solution at point i and time n depends only on the value along the characteristic x - ut = xi - utn according to (6. It is clear that, if this line does not fall inside the domain of dependence, there is trouble, for an attempt is made to determine a value from an irrelevant set of other values. It is therefore necessary that the characteristic line passing through (i, n) be included in the domain of numerical dependence. Except for the undesirable spurious mode, the leapfrog scheme has desirable features, because it is stable for C 1, conserves variance for sufficiently small time steps, and leads to the correct dispersion relation for well-resolved spatial scales. The odd behavior can be explained: In terms of Fourier modes, the solution consists of a series of sine/cosine signals of different wavelength, each of which by virtue of the numerical dispersion relation (6. This also explains the unphysical appearance of both negative values and values in excess of the initial maximum. The cause of the poor performance of the leapfrog scheme is evident: the actual integration should be performed using upstream information exclusively whereas the scheme uses a central average that disregards the origin of the information. To remedy the situation, we now try to take into account the directional information of advection and introduce the so-called upwind or donor cell scheme. A simple Euler scheme over a single time step t is chosen, and fluxes are integrated over this time interval. The essence of this scheme is to calculate the inflow based solely on the average value across the grid cell from where the flow arrives (the donor cell). If the characteristic lies outside the numerical domain of dependence (dashed lines), unphysical behavior will be manifested as numerical instability. One initial condition and one upstream boundary condition are sufficient to determine the numerical solution. The energy method considers the sum of squares of c and determines whether it remains bounded over ~ time, providing a sufficient condition for stability. Ideally the signal should be translated without change in shape by 50 grid points, but the solution is characterized by a certain diffusion and a reduction in gradient. Although it is not related to a physical energy, the method derives its the name from its reliance on a quadratic form that bears resemblance with kinetic energy. Methods that prove that a quadratic form is conserved or bounded over time are similar to energy-budget methods used to prove that the energy of a physical system is conserved. The energy method provides only a sufficient stability condition because the upper bounds used in the demonstration do not need to be reached. Testing the upwind scheme on the "top-hat" problem (Figure 6-8), we observe that, unlike leapfrog, the scheme does not create new minima or maxima, but somehow diffuses the distribution by reducing its gradients. The diffusive behavior can be explained by analyzing the modified equation associated with (6. To give a physical interpretation to the equation, the second time derivative should be replaced by a spatial derivative. Taking the derivative of the modified equation with respect to t provides an equation for the second time derivative, which we would like to eliminate, but it involves a cross derivative3. This cross derivative can be obtained by differentiating the modified equation with respect to x. Some algebra ultimately provides 2c ~ 2c ~ = u2 + O t, x2, t2 x2 which can finally be introduced into (6. Up to O t2, x2, therefore, the numerical scheme solves an advection-diffusion equation instead of the pure advection equation, with diffusivity equal to(1 - C) ux/2. For obvious reasons, this is called an artificial diffusion or numerical diffusion. To decide whether this level of artificial diffusion is acceptable or not, we must compare its size to that of physical diffusion.   