Capturing elaborated flow structures and phenomena is required for well-solved numerical flows. The finite difference methods allow simple discretization of mesh and model equations. However, they need simpler meshes, e.g., rectangular. The inverse Lax-Wendroff (ILW) procedure can handle complex geometries for rectangular meshes. High-resolution and high-order methods can capture elaborated flow structures and phenomena. They also have strong mathematical and physical backgrounds, such as positivity-preserving, jump conditions, and wave propagation concepts. We perceive an effort toward direct numerical simulation, for instance, regarding weighted essentially non-oscillatory (WENO) schemes. Thus, we propose to solve a challenging engineering application without turbulence models. We aim to verify and validate recent high-resolution and high-order methods. To check the solver accuracy, we solved vortex and Couette flows. Then, we solved inviscid and viscous nozzle flows for a conical profile. We employed the finite difference method, positivity-preserving Lax-Friedrichs splitting, high-resolution viscous terms discretization, fifth-order multi-resolution WENO, ILW, and third-order strong stability preserving Runge-Kutta. We showed the solver is high-order and captured elaborated flow structures and phenomena. One can see oblique shocks in both nozzle flows. In the viscous flow, we also captured a free-shock separation, recirculation, entrainment region, Mach disk, and the diamond-shaped pattern of nozzle flows.

Solid and fluid mechanics problems often involve the computation of eigenvalues and eigenvectors. One has short algebraic expressions for the Euler equations and the perfect gas equation of state. However, algebraic expressions for solid mechanics models can become substantial. Computing the eigenvalues and eigenvectors algebraically for different models can become cumbersome. Also, larger formulae may result in increased overall computational time. Internal energy plays a significant role in modeling the medium, and thermodynamic consistency is indispensable. The models should correctly capture heat transfer, mechanical response, and fluid flow, e.g., in mechanocaloric and bone remodeling (BR) problems. In mechanocaloric problems, temperature variation occurs because of load and unload stresses. In BR problems, several mathematical models were developed for modeling its behavior. Flexibility for testing different models is an essential requirement. Hence, our goal is to propose and test approximated eigensystem and singular value (SVD) decompositions for continuum mechanics problems. We tested the resolution and accuracy of our propositions in benchmarking CFD problems. For the solid mechanics, we solved a few classical problems, verified the accuracy, and solved mechanocaloric tests and bar under tension. Although more expensive, the approximated eigensystem decomposition is more flexible. It is high-order and high-resolution, can capture heat transfer, fluid flow, discontinuities, and mechanical response, and can model real-world applications. The SVD is a viable possibility and can be used to adjust the computational time of severe situations, but it needs to be improved.

In this article, we propose a modified convex combination of the polynomial reconstructions of odd-order WENO schemes to maintain the central substencil prevalence over the lateral ones in all parts of the solution. New "centered" versions of the classical WENO-Z and its less dissipative counterpart, WENO-Z+, are defined through very simple modifications of the classical nonlinear weights and show significantly superior numerical properties; for instance, a well-known dispersion error for long-term runs is fixed, along with decreased dissipation and better shock-capturing abilities. Moreover, the proposed centered version of WENO-Z+ has no ad-hoc parameters and no dependence on the powers of the grid size. All the new schemes are thoroughly analyzed concerning convergence at critical points, adding to the discussion on the relevance of such convergence to the numerical simulation of typical hyperbolic conservation laws problems. Nonlinear spectral analysis confirms the enhancement achieved by the new schemes over the standard ones.

This article presents the Brazilian private insurance market’s actuarial life tables, BR- EMS 2021. Using Bayesian inference on the parameters of the Heligman- Pollard law of mortality and data from 23 insurance groups over 15 years, totaling 3.5 billion registers, the data were corrected through a two hidden-layer neural network. The resulting tables show that the insured population exhibits lower mortality rates than the general Brazilian population, even lower than the national populations of well-developed countries such as the USA. Moreover, besides the expected gender gap in mortality rates, there is a clear distance between the death and survivorship insurance coverage groups. Likewise, the insured population characteristics mitigate well-known regional structural discrepancies in the Brazilian population, indicating that being part of the selected population of insured individuals is thus associated with a more effective protection against death than other outstanding factors such as geographic region of residence.

WENO schemes have high-resolution and great capability of capturing unsteady and non-linear phenomena. These phenomena occur in elaborated flows, such as cascade and external aerodynamic flows. Thus, high-resolution methods play an important role in modern CFD. Easiness and effectiveness are reached when implementing these methods in rectangular meshes. However, one has the non-boundary-conforming issue. To overcome it the ILW procedure can be employed. We showed that the ILW has a similar behavior to the far-field boundary treatment. Then, we proposed and tested a new ILW outflow boundary treatment and an WENO-type extrapolation. We employed the finite difference method with the positivity-preserving Lax–Friedrichs splitting, high-resolution viscous terms discretization, multi-resolution WENO scheme, and third-order strong stability preserving Runge–Kutta time discretization. We tested the proposed methods in a smooth test case, supersonic flow past a cylinder, NACA 0012 nitrogen flows with 0 and 12° angle of attack, and a NACA 9520 cascade airflow. The proposed methods have high order and resolution, allowed domain size reduction, and captured unsteady and non-linear phenomena. Although preliminary, the NACA 0012 and 9520 flows showed the methods are promising. Common issues with other methods are also reported and further improvements are needed. For example, local mesh refinement, high-order wall boundary treatment, and turbulence modeling.

Robust numerical methods for CFD applications, such as WENO schemes, quickly evolved in the past few decades. Together with the Inverse Lax–Wendroff (ILW) procedure, WENO ideas were also applied in the boundary treatment. Those methods are known for their high-resolution property, i.e., good representation of nonlinear phenomena, which is an important property in solving challenging engineering problems. In light of that, the objective of this work is to present a review of well-established high-resolution numerical methods to solve the Euler equations and adapt the Navier–Stokes viscous terms discretization and boundary treatment. To test the modifications, we employed the positivity-preserving Lax–Friedrichs splitting, multi-resolution WENO scheme, third-order strong stability preserving Runge–Kutta time discretization, and ILW boundary treatment. The first problems were simple flows with analytical solutions for accuracy tests. We also tested the accuracy with nontrivial phenomena in the vortex flow. Oblique shock and complicated flow structures were captured in the Rayleigh–Taylor instability and flow past a cylinder. We showed the discretization and boundary treatment can handle non-constant viscosity, are high-order, high-resolution, and behave similarly to the well-established numerical methods. Furthermore, the methods discussed here can preserve symmetry and no approximations regarding the boundary layer were made. Therefore, the discretization and boundary treatment can be considered when solving direct numerical simulations.

When solving CFD problems, the solver, or the numerical code, plays an important role. Depending on the phenomena and problem domain, designing such numerical codes can be hard work. One strategy is to start with simple problems and construct the code as building blocks. The purpose of this work is to provide a detailed review of the theory to compute analytical and exact solutions, and recent numerical methods to construct a code to solve compressible and inviscid fluid flows with high-resolution, arbitrary domains, non-linear phenomena, and on rectangular meshes. We also propose a modification to the inverse Lax–Wendroff procedure solid wall treatment and two-dimensional WENO-type extrapolation stencil selection and weights to handle more generic situations. To test our modifications, we use the finite difference method, Lax–Friedrichs splitting, WENO-Z+ scheme, and third-order strong stability preserving Runge-Kutta time discretization. Our first problem is a simple one-dimensional transient problem with periodic boundary conditions, which is useful for constructing the core solver. Then, we move to the one-dimensional Rayleigh flow, which can handle flows with heat exchange and requires more detailed boundary treatment. The next problem is the quasi-one-dimensional nozzle flow with and without shock, where the boundary treatment needs a few adjustments. The first two-dimensional problem is the Ringleb flow, and despite being smooth, it has a curved wall as the left boundary. Finally, the last problem is a two-dimensional conical flow, which presents an oblique shock and an inclined straight line wall being the cone surface. We show that the designed accuracy is being reached for smooth problems, that high-resolution is being attained for non-smooth problems, and that our modifications produce similar results while providing a more generic way to treat solid walls.

Richardson extrapolation is a powerful approach for reducing spatial discretization errors and increasing, in this way, the accuracy of the computed solution obtained by use of many numerical methods for solving different scientific and engineering problems. This approach has been used in a variety of computational fluid dynamics problems to reduce numerical errors, but its use has been restricted mainly to the computation of incompressible fluid flows and on grids with coincident nodes. In this work we present a completed repeated Richardson extrapolation (CRRE) procedure for a more generic type of grid not necessarily with coincident nodes, and test it on compressible fluid flows. Three tests are performed for one-dimensional and quasi-one-dimensional Euler equations: (i) Rayleigh flow, (ii) isentropic flow, and (iii) adiabatic flow through a nozzle. The last test involves a normal shock wave. To build a simple solver, these problems are solved by a first-order upwind-type finite difference method as the base scheme. The normal shock wave problem is also solved with a high-order weighted essentially nonoscillatory (WENO) scheme to compare it with the CRRE procedure. The procedure we propose can increase the achieved accuracy and significantly decrease the magnitude of the spatial error in all three tests. Its performance is best demonstrated in the Rayleigh flow test, where the spatial discretization error is reduced by seven orders of magnitude and the achieved accuracy is increased from 0.998 to 6.62 on a grid with 10,240 nodes. Similar performance is observed for isentropic flow, for which the spatial discretization error is reduced by nine orders of magnitude and the achieved accuracy is increased from 0.996 to 6.73 on a grid with 10,240 nodes. Finally, in adiabatic flow with a normal shock wave, the procedure can reduce the spatial discretization error both upstream and downstream of the shock However, the more expensive high-order WENO scheme results in errors of lower magnitude upstream of the shock and a sharper shock transition for this shocked test case. (C) 2019 Elsevier Inc. All rights reserved.

Nozzle ows have many applications in engineering.Therefore, numerical improvements in such problems are always relevant.Following these lines, we investigated how the WENO (Weighted Essentially Nonoscillatory) scheme deals with shocked nozzle ows.The inviscid ow is modelled via the Euler equations and the numerical ndings are compared with the classical Mac-Cormack scheme.Results show good agreement between the compared solutions.1.

The WENO-Z scheme is known to achieve less dissipative results than the classical WENO scheme, especially in problems involving both shocks and smooth structures. In Acker et al. (J Comput Phys 313:726–753, 2016), the cause of the improved results of WENO-Z was shown to be its comparatively higher weights on less-smooth substencils. This knowledge was exploited to develop the fifth-order WENO-Z+ scheme, which generalizes WENO-Z by including an extra term for increasing the weights of less-smooth substencils even further. The new scheme WENO-Z+ was shown to achieve even better results than WENO-Z, while keeping the same numerical robustness. In this study, the third- and seventh-order versions of the WENO-Z+ scheme are presented and discussed. The preliminary numerical results make evident that the approach used by WENO-Z+ is also sound for orders other than 5.

This paper describes the construction of the BR-EMS 2015 mortality tables for the Brazilian insured population. The tables were based on data collected from insurance companies which represent about 80 per cent of the Brazilian insurance market, and they are updates of their previous versions, BR-EMS 2010, which have been the first mortality tables built with Brazilian market experience. Additional data from government sources was used to improve the information of the companies’ databases. The mortality rates of the population under risk products (death coverage) are remarkably different than those under savings products (survivorship coverage); as such, four different mortality tables are constructed, separating the population by sex as well as the type of insurance coverage. A straight comparison between the BR-EMS 2015 tables with the statistics of the general Brazilian population shows a striking difference on life expectancies. The BR-EMS 2015 tables are also compared with other life tables.

In this article, we show that for a WENO scheme to improve the numerical resolution of smooth waves, increasing to some extent the contribution of the substencils where the solution is less smooth is much more important than improving the accuracy at critical points. WENO-Z, for instance, achieved less dissipative results than classical WENO through the use of a high-order global smoothness measurement, τ, which increased the weights of less-smooth substencils. This time, we present a way of further increasing the relevance of less-smooth substencils by adding a new term to the WENO-Z weights that uses information which is already available in its formula. The improved scheme attains much better resolution at the smooth parts of the solution, while keeping the same numerical stability of the original WENO-Z at shocks and discontinuities.

This paper describes the construction of the BR-EMS 2015 mortality tables for the Brazilian insured population. The tables were based on data collected from insurance companies which represent about 80 per cent of the Brazilian insurance market, and they are updates of their previous versions, BR-EMS 2010, which have been the first mortality tables built with Brazilian market experience. Additional data from government sources was used to improve the information of the companies’ databases. The mortality rates of the population under risk products (death coverage) are remarkably different than those under savings products (survivorship coverage); as such, four different mortality tables are constructed, separating the population by sex as well as the type of insurance coverage. A straight comparison between the BR-EMS 2015 tables with the statistics of the general Brazilian population shows a striking difference on life expectancies. The BR-EMS 2015 tables are also compared with other life tables.

In the reconstruction step of (2r-1) order weighted essentially non-oscillatory conservative finite difference schemes (WENO) for solving hyperbolic conservation laws, nonlinear weights αk and ωk, such as the WENO-JS weights by Jiang et al. and the WENO-Z weights by Borges et al., are designed to recover the formal (2r-1) order (optimal order) of the upwinded central finite difference scheme when the solution is sufficiently smooth. The smoothness of the solution is determined by the lower order local smoothness indicators βk in each substencil. These nonlinear weight formulations share two important free parameters in common: the power p, which controls the amount of numerical dissipation, and the sensitivity ε, which is added to βk to avoid a division by zero in the denominator of αk. However, ε also plays a role affecting the order of accuracy of WENO schemes, especially in the presence of critical points. It was recently shown that, for any design order (2r-1), ε should be of Ω(Δx2) (Ω(Δxm) means that ε⩾CΔxm for some C independent of Δx, as Δx→0) for the WENO-JS scheme to achieve the optimal order, regardless of critical points. In this paper, we derive an alternative proof of the sufficient condition using special properties of βk. Moreover, it is unknown if the WENO-Z scheme should obey the same condition on ε. Here, using same special properties of βk, we prove that in fact the optimal order of the WENO-Z scheme can be guaranteed with a much weaker condition ε=Ω(Δxm), where m(r,p)⩾2 is the optimal sensitivity order, regardless of critical points. Both theoretical results are confirmed numerically on smooth functions with arbitrary order of critical points. This is a highly desirable feature, as illustrated with the Lax problem and the Mach 3 shock-density wave interaction of one dimensional Euler equations, for a smaller ε allows a better essentially non-oscillatory shock capturing as it does not over-dominate over the size of βk. We also show that numerical oscillations can be further attenuated by increasing the power parameter 2⩽p⩽r-1, at the cost of increased numerical dissipation. Compact formulas of βk for WENO schemes are also presented.

In this work we investigate the spectral signature of Navier-Stokes-Voigt (NSV) viscoelastic fluid flows by employing numerical simulations of a singular dyadic shell model. Our results clearly show that as the relaxation time is increased above a threshold, the inertial range is reduced, conserving part of the large-scale statistics. These results differ drastically from the two power-law scenarios observed in a previous work, where the NSV model was studied via Sabra shell model simulations instead. We also show that the additional elastic term regularizes the singular dyadic model, which is the main reason behind this reduction of degrees of freedom. The results of this work aim at proposing the NSV regularization as a sub-grid model.

In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.