Numerical Issues truncation error

The successful numerical solution of differential equations requires attention to two issues associated with error control through time step selection. One is accuracy and the other is stability. Accuracy requires a time step that is sufficiently small so that the numerical solution is close to the true solution. Numerical methods usually measure the accuracy in terms of the local truncation error, which depends on the details of the method and the time step. Stability requires a time step that is sufficiently small that numerical errors are damped, and not amplified. A problem is called stiff when the time step required to maintain stability is much smaller than that that would be required to deliver accuracy, if stability were not an issue. Generally speaking, implicit methods have very much better stability properties than explicit methods, and thus are much better suited to solving stiff problems. Since most chemical kinetic problems are stiff, implicit methods are usually the method of choice. 

In an implementation of the DKH protocol for actual calculations, exact decoupling is reached if the truncation error 0(1/"+ ) is of the order of the arithmetic precision of the computer, or at least negligible either with respect to the physical or chemical issues xmder investigation or with respect to additional methodological approximations such as the finite size of a one-particle basis set. Given a molecular system and a desired numerical accuracy t, we may determine a cut-off order riopt of the DKH expansion prior to the calculation, beyond which no further numerical improvement (relevant for t) will be achieved. 

We caution that issues beyond accuracy are involved. As noted in Chapter 13, the computed diffusivity is not the physical diffusivity k, but a combination of that plus numerical diffusion due to truncation errors. 

Here we will reconsider the reservoir in Example 9-5. There, the well was in a sense newly drilled and was allowed to attain its maximum flow rate in time, under pressure-pressure boundary conditions. Here, we will consider the reverse situation. The reservoir is assumed to be flowing at steady state initially in accordance with Figures 10-lh,i. Suddenly at t = 0+ hours, the well is completely shut in, so that Qw = 0 cubic feet/hour. Thus, we expect I3v to increase with time, and the reservoir to achieve everywhere the 900 psi set at the farfield. But the problems here and in Example 9-5 are not identical, with the direction of time simply reversed. Numerically, we have different truncation and cumulative error histories, and stability issues arising from contrasting initial-boundary conditions are unlike. 

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