Monotonicity or boundedness properties (e.g. strong-stability-preserving, total-variation-diminishing or total-variation-boundedness properties) for linear multistep methods (LMMs) can be guaranteed by imposing step-size restrictions on the methods. To describe these restrictions, one introduces the concept of step-size coefficient for monotonicity (SCM, also referred to as the strong-stability-preserving (SSP) coefficient) and its generalization, the step-size coefficient for boundedness (SCB coefficient). A LMM with larger SCM or SCB is more efficient. The computation of the maximum SCM for a particular LMM is straightforward, however, it is more challenging to decide whether a SCB exists, or determine if a given number is a SCB. Based on some recent theorems in the literature we present methods to find the exact optimal SCB for a LMM. As an illustration, we consider SCBs in the BDF, extrapolated BDF, and Adams--Bashforth families.
Recently developed combustion models based on detailed reaction mechanisms may provide good results. The next question is how accurate the calculated results are, in other words, what is their uncertainty. The main source of uncertainty of the simulation results is the uncertainty of the chemical kinetic parameters. These parameters describe the temperature and pressure dependence of the rate coefficients. A first guess of the uncertainty of these parameters (“prior uncertainty”) may be based on directly measured rate coefficients and theoretical calculations. The temperature dependent uncertainty bands of the rate coefficients can be converted to the uncertainty domains of the Arrhenius parameters. Detailed reaction mechanisms can be developed and checked using the results of indirect measurements, like ignition delay time measurements in shock tubes and rapid compression machines, laminar burning velocity measurements, and concentration profile determinations in various reactors. At the optimization of detailed reaction mechanisms, all important rate parameters are fitted in one step within their domain of prior uncertainty using a global optimization method, taking into account all relevant indirect and direct measurements. The results of optimization are the set of the best estimated values of the important parameters that can be obtained from the experimental data considered, and also their covariance matrix, which is a good representation of their joint uncertainty. This posterior uncertainty domain is usually much smaller than the prior uncertainty one. The experimental data are suggested to be stored in ReSpecTh Kinetics Data Format files, allowing software independent and permanent storage. A series of software tools have been developed which allow the determination of prior and posterior uncertainty limits by processing ReSpecTh Kinetics Data Format files.
We present a novel computational tool for establishing the existence of periodic solutions of piecewise-linear delay differential equations. The method is based on the Chebyshev expansions of solutions. As an illustration, we use a piecewise-linear approximation of the celebrated Mackey-Glass equation.