Reduced Chemistry Models Satisfying Error Bounds Over Ranges

Constructing Reduced Chemistry Models Satisfying Error Bounds Over Ranges 

The algebraic equations in the chemistry operator co can be much more complicated than the equations usually used for the transport operator 0, but it is still possible to bound the error introduced by approximating co, by using interval analysis. Interval analysis is a branch of mathematics that considers how mathematical operations affect intervals (ranges) [ Tlow, Thigh] rather than single points Y. For error control, one wants to rigorously bound the error 

Interval analysis is trickier than ordinary mathematics, since in interval analysis some arithmetic operations give outputs which are overestimates of the true range. In other words, if a function f x) takes on values between /min and /max for input values of. v in the interval [.V ow, xhigh], interval analysis may give you an output range f mJ, where/ x /max and/or f min fmia. 

As a result, interval analysis tends to overestimate error bounds. However, there are clever ways to reduce this overestimation, for example by appropriately grouping terms. One of the best and most computationally efficient ways to minimize overestimation of the bounds on the approximation error is to replace terms f Y) in w(Y) by their Taylor models, e.g. the first-order Taylor model is given by Eq. (14)  

In ordinary arithmetic one normally neglects the final term, making this an approximation rather than a rigorous equality, since all one knows is the interval in which lies  

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