Marching backwards

The marching-ahead equation becomes a marching-backward equation. The method is called reverse shooting. The procedure is to guess aj = aout and then to set aj / = aj. The index j in Equation (9.30) begins at J — 2 and is decremented by 1 until / = 0 is reached. The reaction rate continues to be evaluated at the central, /th point. The test condition is whether ain is correct when calculated using the inlet boundary condition... 

Jeffries, T., Lindblad, P. (2009). We march backwards into the future. Current Opinion in Biotechnology, 20, 255—256. 

The marching equation for reverse shooting. Equation (9.24), was developed using a first-order, backward difference approximation for dajdz, even though a second-order approximation was necessary for (faldz. Since the locations j—l, j, and j+ are involved an5rway, would it not be better to use a second-order, central difference approximation for dajdz ... 

MJBBE Marching jury backward beam equation... 

Atmadj a J (2001) The marching-jury backward beam equation method and its application to backtracking non-reactive plumes in groundwater. PhD Dissertation, Columbia University, p 121... 

To compute unsteady flows, the time derivative terms in the governing equations need to be discretized. The major difference in the space and time co-ordinates lies in the direction of influence. In unsteady flows, there is no backward influence. The governing equations for unsteady flows are, therefore, parabolic in time. Therefore, essentially all the numerical methods advance in time, in a step-by-step or marching approach. These methods are very similar to those applied for initial value problems (IVPs) of ordinary differential equations. In this section, some of the methods widely used in the context of the finite volume method are discussed. 

The backward and forward movements of Kuroshio waters (also named the Philippine Sea waters), which pass through the Luzon Strait at the intermediate layer, can be well confined by the 34.6%o isohaline. The maximum (minimum) intrusion of the salinity tongue arises in June (October). The warm, saltier Kuroshio water enters the SCS through the Luzon Strait from October to March, whose maximum (minimum) intrusion arises in February (September) with a volume transport of about 13.7 Sv (1.4 Sv), and the annual mean volume transport is about 6.5 Sv (Chu and Li, 2000). The saltiest North Pacific water (NPW) enters the SCS all the year round with a strong intrusion present in winter and summer, and a weak intrusion present... 

For years now we ve been leaning over backwards to accommodate the teenagers. Accepting meekly on the radio and television it is THEIR music which monopolizes the air. That in our shops it is THEIR fads which will dictate our dress styles. .. we have watched them patiently through the wilder excesses of their ban the bomb marches. Smiled indulgently as they ve wrecked our cinemas during their rock and roll films. .. But when they start dragging elderly women around the streets. .. etc. 

The related fast marching methods (Sethian 1999) have also been applied to image segmentation. These methods are faster, but are designed for problems in which the speed function never changes sign, so that the front is always moving forward or backward. 

The measured ammonia and ammonium concentrations at the Finnish EMEP measurement stations of Virolahti and Ahtari show a maximum on March 21, 1989 this time corresponds to computed arrival times. However, such maximum values are not uncommon in springtime. A detailed analysis of backward trajectories, computed for these measurement stations, and the measured concentrations do not show conclusively that the measured maximum values would have been caused by the accident (Kukkonen et al., 1993). Most of the NH due to the accident may have escaped the available measurement stations. 

Numerical analysis. Equation 21-76 is conveniently solved, again using finite difference time-marching schemes. We always central difference our first and second-order space derivatives, while backward differencing in time, with respect to the nodal point (ri,tn). Furthermore, we will evaluate all nonlinear saturation-dependent coefficients at their previous values in time. This leads to... 

Coulaloglou and Tavlarides (1977) 1977 Iterative technique backward marching from largest size increment Direct Used phenomenological models to develop breakage and coalescence rate and distribution functions. Used these to predict the DSD, which compared favorably with CSTR data. 

The backward differencing method requires the solution of 7+1 simultaneous equations to find the radial temperature profile. It is semi-implicit since the solution is still marched-ahead in the axial direction. Fully implicit schemes exist where (7-I- l)(7-l-1) equations are solved simultaneously, one for each grid point in the total system. Fully implicit schemes may be used for problems where axial diffusion or conduction is important so that second derivatives in the axial direction, or 9 r/9z, must be retained in the partial differential equa-... 

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