No abstract

This work proposes a new spatial reconstruction scheme in finite volume frameworks. Different from long-lasting reconstruction processes which employ high order polynomials enforced with some carefully designed limiting projections to seek stable solutions around discontinuities, the current discretized scheme employs THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive sharpness to solve both smooth and discontinuous solutions. Due to the essentially monotone and bounded properties of THINC function, difficulties to solve sharp discontinuous solutions and complexities associated with designing limiting projections can be prevented. A new simplified BVD (Boundary Variations Diminishing) algorithm, so-called adaptive THINC-BVD, is devised to reduce numerical dissipations through minimizing the total boundary variations for each cell. Verified through numerical tests, the present method is able to capture both smooth and discontinuous solutions in Euler equations for compressible gas dynamics with excellent solution quality competitive to other existing schemes. More profoundly, it provides an accurate and reliable solver for a class of reactive compressible gas flows with stiff source terms, such as the gaseous detonation waves, which are quite challenging to other high-resolution schemes. The stiff C-J detonation benchmark test reveals that the adaptive THINC-BVD scheme can accurately capture the reacting front of the gaseous detonation, while the WENO scheme with the same grid resolution generates unacceptable results. Owing also to its algorithmic simplicity, the proposed method can become as a practical and promising numerical solver for compressible gas dynamics, particularly for simulations involving strong discontinuities and reacting fronts with stiff source term.

In this study, a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called P n T m − BVD (polynomial of n-degree and THINC function of m-level reconstruction based on BVD algorithm), are designed by employing high-order linear-weight polynomials and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates.The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using linear-weight polynomials. The analysis of the spectral property and accuracy tests show that the new reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order linear-weight polynomials for smooth solution over all wave numbers and realizes n+1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation.With outperforming results and simplicity in algorithm, the new reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.

The purpose of this work is to build a general framework to reconstruct the underlying fields within a finite volume (FV) scheme solving a hyperbolic system of PDEs (Partial Differential Equations). In an FV context, the data are piecewise constants per computational cell and the physical fields are reconstructed taking into account neighbor cell values. These reconstructions are further used to evaluate the physical states usually used to feed a Riemann solver which computes the associated numerical fluxes. The physical field reconstructions must obey some properties linked to the system of PDEs such as the positivity, but also some numerically based ones, like an essentially nonoscillatory behavior. Moreover, the reconstructions should be highly accurate for smooth flows and robust/stable for discontinuous solutions. To ensure a solution property preserving reconstruction, we introduce a methodology to blend high-/low-order polynomials and hyperbolic tangent reconstructions. A boundary variation diminishing algorithm is employed to select the best reconstruction in each cell. An a posteriori MOOD detection procedure is employed to ensure the positivity by recomputing the rare problematic cells using a robust first-order FV scheme. We illustrate the performance of the proposed scheme via the numerical simulations for some benchmark tests which involve vacuum or near vacuum states, strong discontinuities, and also smooth flows. The proposed scheme maintains high accuracy on smooth profile, preserves the positivity and can eliminate the oscillations in the vicinity of discontinuities while maintaining sharper discontinuity with superior solution quality compared to classical highly accurate FV schemes. K E Y W O R D Sfinite volume, hyperbolic system of PDEs, MOOD, multi-stage-BVD, positivity-preserving, THINC Int J Numer Meth Fluids. 2020;92:603-634.wileyonlinelibrary.com/journal/fld

We propose an estimation method of population moments or parameters in “biased sampling data” in which for some units of data, not only the variables of interest but also the covariates have missing observations, and the proportion of “missingness” is unknown. We use auxiliary information such as the distribution of covariates or their moments in random sampling data in order to correct the bias. Moreover, with additional assumptions, we can correct the bias even if we have only the moment information of covariates. The main contribution of this paper is the development of a doubly robust‐type estimator for biased sampling data. This method provides a consistent estimator if either the regression function or the assignment mechanism is correctly specified. We prove the consistency and semi‐parametric efficiency of the doubly robust estimator. Both the simulation and empirical application results demonstrate that the proposed estimation method is more robust than existing methods.