Method right elimination

The algorithm presented below as the sequence of applied formulae is called the right elimination method and is showing the gateway for the... 

Returning to the right elimination method, we show that the conditions ai < 1 guarantee that the error 6yi y = — j/j+i arising when... 

The left elimination method. The counter elimination method. Still using the framework of the right elimination method (formulae (10)-(15)) in reverse order, we obtain through such an analysis the computational formulae of the left elimination method ... 

Joint use of the left and right elimination methods refers to the counter elimination method. The essence of this method is to consider a fixed inner... 

The computational formulae of the right elimination method help derive estimate (38) ... 

Stability of the elimination method. Let us stress that the conditions Ci cti Ai 0 and 1 — aN 0 cannot be excluded or relaxed during the course of the right elimination method. Just for this reason the restrictions on coefficients for well-posed and stability conditions needs investigation. 

This algorithm is stable because the solutions of (79) are found by the right elimination method being stable under conditions (77) with the denominator 1 — Tw+i > O Indeed, it follows from (77)-(78) that... 

Fig. 1.1 Illustration of the projection method to eliminate undesired components of a vector. Le/i a is projected on the x-axis Middle the projection (P a) is subtracted from a Right The result of the operation is the y-component of a...

Fig. 5.11. Left panel Schematic of the pulsed valve discharge modulation source. A 100 kHz square wave discharge 1 kV, lA) is strongly confined in the 1 mmxSOO /am X 4 cm region behind the slit expansion jaws. This yields slabs of spatially modulated jet-cooled radicals and molecular ions for detection via direct IR laser absorption and lock-in detection methods. Right panel Upper trace direct absorption spectra with conventional discharge modulation in a cw discharge source but without lock-in detection Lower trace spectra with frequency modulated discharge and lock-in detection, revealing substantial elimination of radical precursor and improves the absorption sensitivity to the near shot-noise level.

Revisions of the continuous-flow method have been made to allow observations along the length of the flow tube rather than at right angles.5 This method, fast continuous flow, eliminates the dead time during which the reaction cannot be observed. Kinetic data can be extracted to a time resolution of nearly 10 p,s, but the mathematics is more complicated in this limit, because the mixing and chemical reaction occur on the same time scale. Rate constants nearly as large as the diffusion-controlled value have been determined in favorable cases.6... 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

---