MultiD Analyses (Sweden)

The two decades old high order central differencing via entropy splitting and summation-by-parts (SBP) difference boundary closure of Olsson and Oliger (Energy and maximum norm estimates for nonlinear conservation laws, 1994), Gerritsen and Olsson (J Comput Phys 129:245–262, 1996) and Yee et al. (J Comput Phys 162:33–81, 2000) is revisited. The objective of this paper is to prove for the first time that the entropy split methods based on physical entropies are entropy stable methods for central differencing with SBP operators for both periodic and non-periodic boundary conditions for nonlinear Euler equations. Standard high order spatial central differencing as well as high order central spatial dispersion relation preserving spatial differencing is part of the entropy stable methodology framework. The proof is to replace the spatial derivatives by SBP difference operators in the entropy split form of the equations using the physical entropy of the Euler equations. The numerical boundary closure follows directly from the SBP operator. No additional numerical boundary procedure is required. In contrast, Tadmor-type entropy conserving methods (Acta Numer 12:451–512, 2003) using mathematical entropies and more recently the methods in Winters and Gassner (J Comput Phys 304:72–108, 2016), do not naturally come with a numerical boundary closure and a generalized SBP operator has to be developed (Roanocha in Generalized summation-by-parts operators and variable coefficients, 2018. arXiv:1705.10541v2 [math.NA]). Long time integration of 2D and 3D test cases is included to show the comparison of this efficient entropy stable method with the Tadmor-type of entropy conservative methods. Studies also include the comparison among the three skew-symmetric splittings on their nonlinear stability and accuracy performance without added numerical dissipations for smooth flows. These are, namely, entropy splitting, Ducros et al. splitting and the Kennedy and Grubber splitting.

To compare health-related quality of life in 2 different populations with chronic pain: patients with fibromyalgia and patients with refractory angina pectoris. Previous separate studies have indicated that these patient groups report different impacts of pain on health-related quality of life. The Short-Form 36 was used to assess health- related quality of life. In order to adjust for age and gender differences between the groups, both patient groups were compared with age- and gender-matched normative controls. The difference in health-related quality of life between the 2 patient groups was assessed by transforming the Short-Form 36 subscale scores to a z-score. The patients with fibromyalgia (n = 203) reported poorer health-related quality of life in all the subscale scores of Short-Form 36 (p < 0.05-0.0001) than the patients with refractory angina (n = 146) when both groups were compared with their corresponding normal population (z-score). Patients with fibromyalgia experience greater impairment in health-related quality of life compared with the normal population than do patients with refractory angina pectoris, despite the fact that the latter have a potentially life-threatening disease. The great impairment in health- related quality of life in patients with fibromyalgia should be taken into consideration when planning rehabilitation.