Essentially optimal explicit Runge–Kutta methods with application to hyperbolic–parabolic equations

Optimal explicit Runge–Kutta methods consider more stages in order to include a particular spectrum in their stability domain and thus reduce time-step restrictions. This idea, so far used mostly for real-line spectra, is generalized to more general spectra in the form of a thin region. In thin regions the eigenvalues may extend away from the real axis into the imaginary plane. We give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these polynomials for various cases. Semi-discretizations of hyperbolic–parabolic equations are a relevant application which exhibit a thin region spectrum. As a model, linear, scalar advection–diffusion is investigated. The second-order-stabilized explicit Runge–Kutta methods derived from the stability polynomials are applied to advection–diffusion and compressible, viscous fluid dynamics in numerical experiments. Due to the stabilization the time step can be controlled solely from the hyperbolic CFL condition even in the presence of viscous fluxes.     

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A Titration Microcalorimetric Study of the Effects of Halide Counterions on Vesicle-Forming Aggregation in Aqueous Solution of Branched-Chain Alkylpyridinium Surfactants

Titration microcalorimetry is used to study the influences of iodide, bromide, and chloride counterions on the aggregation of vesicle-forming 1-methyl-4-(2-pentylheptyl)pyridinium halide surfactants. Formation of vesicles by these surfactants was characterised using transmission electron microscopy. When the counterion is changed at 303 K through the series iodide, bromide, to chloride, the critical vesicular concentration (cvc) increases and the enthalpy of vesicle formation changes from exo- to endothermic. With increase in temperature to 333 K, vesicle formation becomes strongly exothermic. Increasing the temperature leads to a decrease in enthalpy and entropy of vesicle formation for all three surfactants. However the standard Gibbs energy for vesicle formation is, perhaps surprisingly, largely unaffected by an increase in temperature, as a consequence of a compensating change in both standard entropy and standard enthalpy of vesicle formation. Interestingly, standard isobaric heat capacities of vesicle formation are negative, large in magnitude but not strikingly dependent on the counterion. We conclude that the driving force for vesicle formation can be understood in terms of overlap of the thermally labile hydrophobic hydration shells of the alkyl chains. Copyright 2000 Academic Press.