We develop a general framework for solving scalar conservation laws using finite volume weighted essentially non oscillatory (WENO) techniques on general computational meshes in multiple space dimensions. We address two fundamental issues. First, polynomial approximations on general stencils of mesh cells can be of poor quality, even for what appear to be geometrically nice stencils. We present a robust and efficient procedure for producing accurate stencil polynomial approximations. Bad stencils are identified by considering the condition number of the linear system used to define the stencil polynomial. Second, we develop a novel and efficient finite volume, multilevel WENO (ML-WENO) reconstruction that is flexible enough to be applied effectively in a variety of settings and with essentially any reasonable set of stencils. It combines stencil polynomial approximations of various degrees with a nonlinear weighting biasing the reconstruction away from both inaccurate oscillatory polynomials of high degree (i.e., those crossing a shock or steep front) and smooth polynomials of low degree, thereby selecting the smooth polynomial(s) of maximal degree of approximation. We conduct numerical tests showing poor quality mesh stencils, the behavior of the reconstruction for both smooth and discontinuous functions, and applications to scalar conservation laws.

Finite volume, weighted essentially non-oscillatory (WENO) schemes require the computation of a smoothness indicator. This can be expensive, especially in multiple space dimensions. We consider the use of the simple smoothness indicator σ ^S= 1/N_S-1∑ _j (u̅_j - u̅_m)^2 , where N_S is the number of mesh elements in the stencil, u̅_j is the local function average over mesh element j, and index m gives the target element. Reconstructions utilizing standard WENO weighting fail with this smoothness indicator. We develop a modification of WENO-Z weighting that gives a reliable and accurate reconstruction of adaptive order, which we denote as SWENOZ-AO. We prove that it attains the order of accuracy of the large stencil polynomial approximation when the solution is smooth, and drops to the order of the small stencil polynomial approximations when there is a jump discontinuity in the solution. Numerical examples in one and two space dimensions on general meshes verify the approximation properties of the reconstruction. They also show it to be about 10 times faster in two space dimensions than reconstructions using the classic smoothness indicator. The new reconstruction is applied to define finite volume schemes to approximate the solution of hyperbolic conservation laws. Numerical tests show results of the same quality as standard WENO schemes using the classic smoothness indicator, but with an overall speedup in the computation time of about 3.5–5 times in 2D tests. Moreover, the computational efficiency (CPU time versus error) is noticeably improved.

This paper develops a framework for finite volume radial basis function (RBF) approximation of a function u on a stencil of mesh cells in multiple dimensions. The theory of existence of the approximation is given. In one dimension, as the cell diameters tend to zero, numerical evidence is given to show that the RBF approximation converges to u to the same order as a polynomial approximation when the RBF is infinitely differentiable. Specific multiquadric RBFs on stencils of 2 and 3 mesh cells are proven to have this convergence property. A two-level RBF based weighted essentially non-oscillatory (WENO) reconstruction with adaptive order (RBF-WENO-AO) is developed. WENO-AO reconstructions use arbitrary linear weights, and so they can be developed easily for RBF approximations, even on nonuniform meshes in multiple dimensions. Following the classical polynomial based WENO, a smoothness indicator is defined for the reconstruction. For one dimension, the convergence theory is given regarding the cases when u is smooth and when u has a discontinuity. These reconstructions are applied to develop finite volume schemes for hyperbolic conservation laws on nonuniform meshes over multiple space dimensions. The focus is on reconstructions based on multiquadric RBFs that are third order when the solution is smooth and second order otherwise, i.e., RBF-WENO-AO(3,2). Numerical examples show that the scheme maintains proper accuracy and achieves the essentially non-oscillatory property when solving hyperbolic conservation laws.

A finite volume approximation of the scalar hyperbolic conservation law or advection-diffusion equation is given. In the context of the method of lines, the space discretization uses weighted essentially non oscillatory (WENO) reconstructions with adaptive order (WENO-AO), and the time evolution uses implicit Runge-Kutta methods. Therefore the timestep may be larger than the CFL timestep. To reduce oscillation in the solution, ideas related to spatially partitioned Runge-Kutta methods are used. An adaptive Runge-Kutta method is developed that blends the L-stable, third order, implicit Radau IIA method with the composite backward Euler method using a weighting procedure inspired from spatial WENO methods. The weighting procedure requires a smoothness indicator, and several possibilities are considered, although one is perhaps seen to be preferred. The overall scheme is proven to maintain third order accuracy when the solution is smooth. When the solution has a discontinuity, the scheme is shown computationally to be third order accurate away from shocks, and to achieve the overall accuracy of the backward Euler method. Numerical examples show that the adaptive Runge-Kutta method reduces oscillations in the solution. Moreover, the resulting scheme is shown to be unconditionally L-stable for smooth solutions to the linear problem. (C) 2020 Elsevier B.V. All rights reserved.

We consider numerical approximation of the degenerate advection-diffusion equation, which is formally parabolic but may exhibit hyperbolic behavior. We develop both explicit and implicit finite volume weighted essentially nonoscillatory (WENO) schemes in multiple space dimensions on non-uniform computational meshes. The diffusion degeneracy is reformulated through the use of the Kirchhoff transformation. Space is discretized using WENO reconstructions with adaptive order (WENO-AO), which have several advantages, including the avoidance of negative linear weights and the ability to handle irregular computational meshes. A special two-stage WENO reconstruction procedure is developed to handle degenerate diffusion. Element averages of the solution are first reconstructed to give point values of the solution, and these point values are in turn used to reconstruct the Kirchhoff transform variable of the diffusive flux. Time is discretized using the method of lines and a Runge-Kutta time integrator. We use Strong Stability Preserving (SSP) Runge-Kutta methods for the explicit schemes, which have a severe parabolically scaled time step restriction to maintain stability. We also develop implicit Runge-Kutta methods. SSP methods are only conditionally stable, so we discuss the use of L-stable Runge-Kutta methods. We present in detail schemes that are third order in both space and time in one and two space dimensions using non-uniform meshes of intervals or quadrilaterals. Efficient implementation is described for computational meshes that are logically rectangular. Through a von Neumann (or Fourier mode) stability analysis, we show that smooth solutions to the linear problem are unconditionally L-stable on uniform computational meshes when using an implicit Radau IIA Runge-Kutta method. Computational results show the ability of the schemes to accurately approximate challenging test problems.

We consider numerical approximation of the degenerate advection-diffusion equation, which is formally parabolic but may exhibit hyperbolic behavior. We develop both explicit and implicit finite volume weighted essentially non-oscillatory (WENO) schemes in multiple space dimensions on non-uniform computational meshes. The diffusion degeneracy is reformulated through the use of the Kirchhoff transformation. Space is discretized using WENO reconstructions with adaptive order (WENO-AO), which have several advantages, including the avoidance of negative linear weights and the ability to handle irregular computational meshes. A special two-stage WENO reconstruction procedure is developed to handle degenerate diffusion. Element averages of the solution are first reconstructed to give point values of the solution, and these point values are in turn used to reconstruct the Kirchhoff transform variable of the diffusive flux. Time is discretized using the method of lines and a Runge-Kutta time integrator. We use Strong Stability Preserving (SSP) Runge-Kutta methods for the explicit schemes, which have a severe parabolically scaled time step restriction to maintain stability. We also develop implicit Runge-Kutta methods. SSP methods are only conditionally stable, so we discuss the use of L-stable Runge-Kutta methods. We present in detail schemes that are third order in both space and time in one and two space dimensions using non-uniform meshes of intervals or quadrilaterals. Efficient implementation is described for computational meshes that are logically rectangular. Through a von Neumann (or Fourier mode) stability analysis, we show that smooth solutions to the linear problem are unconditionally L-stable on uniform computational meshes when using an implicit Radau IIA Runge-Kutta method. Computational results show the ability of the schemes to accurately approximate challenging test problems.

In this paper, we analyze standard weighted essentially nonoscillatory (WENO) reconstructions and multilevel WENO reconstructions with adaptive order (WENO-AO) using both WENO-JS and WENO-Z weighting. We also present a new WENO-AO reconstruction. We give conditions under which the reconstructions achieve optimal order accuracy for both smooth solutions and solutions with discontinuities. The old WENO-AO reconstruction drops to a fixed, base level of approximation when there are discontinuities in the solution, but the new one maintains the accuracy of the largest stencil over which the solution is smooth. Our analysis in the discontinuous case requires that the smoothness indicators do not approach zero as the grid is refined. We provide a condition to ensure this result, but we also show an example where this can fail to occur. That is, we show that WENO reconstructions can fail to maintain the order of approximation of the smallest stencil over which the solution is smooth. We also present numerical results confirming the convergence theory of the old and new WENO-AO reconstructions and compare their performance in solving conservation laws.

We present a new, formally third order, implicit Weighted Essentially Non-Oscillatory (iWENO3) finite volume scheme for solving systems of nonlinear conservation laws. We then generalize it to define an implicit Eulerian---Lagrangian WENO (iEL-WENO) scheme. Implicitness comes from the use of an implicit Runge---Kutta (RK) time integrator. A specially chosen two-stage RK method allows us to drastically simplify the computation of the intermediate RK fluxes, leading to a computationally tractable scheme. The iEL-WENO3 scheme has two main steps. The first accounts for particles being transported within a grid element in a Lagrangian sense along the particle paths. Since this particle velocity is unknown (in a nonlinear problem), a fixed trace velocity v is used. The second step of the scheme accounts for the inaccuracy of the trace velocity v by computing the flux of particles crossing the incorrect tracelines. The CFL condition is relaxed when v is chosen to approximate the characteristic velocity. A new Roe solver for the Euler system is developed to account for the Lagrangian tracings, which could be useful even for explicit EL-WENO schemes. Numerical results show that iEL-WENO3 is both less numerically diffusive and can take on the order of about 2---3 times longer time steps than standard WENO3 for challenging nonlinear problems. An extension is made to the advection---diffusion equation. When advection dominates, the scheme retains its third order accuracy.

We study the electronic structure of the skutterudite Ce3Co4Sn13, which is known to exhibit a charge density wave (CDW) transition, at temperature T-CDW similar to 160 K, coupled to a metal-to-metal transition. We use temperature dependent hard x-ray photoemission spectroscopy (HAXPES) and x-ray absorption spectroscopy (XAS) to investigate the occupied and unoccupied electronic states of Ce3Co4Sn13. The Co 2p and Sn 3p core level spectra show small but finite shifts in binding energy positions across T-CDW while Ce 3d core level spectra do not show any change across the transition. The Ce M-4,M-5-edge XAS spectrum compared with calculations indicate a typical trivalent ionic Ce3+ spectral shape, ruling out Kondo screening in Ce3Co4Sn13. In contrast, the Co L-2,L-3-edge XAS spectrum compared with a calculated spectrum shows evidence for hybridization with neighboring Sn atoms in a trigonal prismatic co-ordination. Temperature dependent XAS across the Co L-2,L-3-edge shows a small shift across T-CDW, consistent with HAXPES results. Detailed XAS measurements as a function of temperature show that the spectral shifts occur with a hysteresis across T-CDW , indicative of a first-order transition. Valence band spectra show a normal Fermi edge above and below T-CDW. The Co 3d states are observed at a binding energy of similar to 2 eV while the Ce 4f states occur as a weak feature within 0.5 eV of the Fermi level. The results suggest an unusual CDW transition coupled to a metal-to-metal transition in Ce3Co4Sn13.

We report a study of the temperature-dependent electrical resistivity, Seebeck coefficient, thermal conductivity, specific heat, and Al-27 nuclear magnetic resonance (NMR) in Heusler-type Ru2TaAl, to shed light on its semimetallic behavior. While the temperature dependence of the electrical resistivity exhibits semiconductor like behavior, the analysis of low-temperature specific heat reveals a residual Fermi-level density of states (DOS). Both observations can be realized by means of a semimetallic scenario with the Fermi energy located in the pseudogap of the electronic DOS. The NMR Knight shift and spin-lattice relaxation rate show activated behavior at higher temperatures, attributing to the thermally excited carriers across a pseudogap in Ru2TaAl. From the first-principles band structure calculations, we further provide a clear picture that an indirect overlap between electron and hole pockets is responsible for the formation of a pseudogap in the vicinity of the Fermi level of Ru2TaAl. In addition, an effort for improving the thermoelectric performance of Ru2TaAl has been made by investigating the thermoelectric properties of Ru1.95Ta1.05Al. We found significant enhancements in the electrical conductivity and Seebeck coefficient and marked reduction in the thermal conductivity via the off-stoichiometric approach. This leads to an increase in the figure-of-merit ZT value from 6.1 x 10(-4) in Ru2TaAl to 3.4 x 10(-3) in Ru1.95Ta1.05Al at room temperature. In this respect, a further improvement of thermoelectric performance based on Ru2TaAl through other off-stoichiometric attempts is highly probable.

We develop a formally high order Eulerian–Lagrangian Weighted Essentially Nonoscillatory (EL‐WENO) finite volume scheme for nonlinear scalar conservation laws that combines ideas of Lagrangian traceline methods with WENO reconstructions. The particles within a grid element are transported in the manner of a standard Eulerian–Lagrangian (or semi‐Lagrangian) scheme using a fixed velocity v . A flux correction computation accounts for particles that cross the v ‐traceline during the time step. If v = 0, the scheme reduces to an almost standard WENO5 scheme. The CFL condition is relaxed when v is chosen to approximate either the characteristic or particle velocity. Excellent numerical results are obtained using relatively long time steps. The v ‐traceback points can fall arbitrarily within the computational grid, and linear WENO weights may not exist for the point. A general WENO technique is described to reconstruct to any order the integral of a smooth function using averages defined over a general, nonuniform computational grid. Moreover, to high accuracy, local averages can also be reconstructed. By re‐averaging the function to a uniform reconstruction grid that includes a point of interest, one can apply a standard WENO reconstruction to obtain a high order point value of the function. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 651–680, 2017

For a nonlinear scalar conservation law in one-space dimension, we develop a locally conservative semi-Lagrangian finite difference scheme based on weighted essentially non-oscillatory reconstructions (SL-WENO). This scheme has the advantages of both WENO and semi-Lagrangian schemes. It is a locally mass conservative finite difference scheme, it is formally high-order accurate in space, it has small time truncation error, and it is essentially non-oscillatory. The scheme is nearly free of a CFL time step stability restriction for linear problems, and it has a relaxed CFL condition for nonlinear problems. The scheme can be considered as an extension of the SL-WENO scheme of Qiu and Shu (2011) [2] developed for linear problems. The new scheme is based on a standard sliding average formulation with the flux function defined using WENO reconstructions of (semi-Lagrangian) characteristic tracings of grid points. To handle nonlinear problems, we use an approximate, locally frozen trace velocity and a flux correction step. A special two-stage WENO reconstruction procedure is developed that is biased to the upstream direction. A Strang splitting algorithm is used for higher-dimensional problems. Numerical results are provided to illustrate the performance of the scheme and verify its formal accuracy. Included are applications to the Vlasov–Poisson and guiding-center models of plasma flow.

Solving multi-dimensional eigenmodes problem for elliptic operator using radial basis functions (RBFs) was proposed by Platte and Driscoll (2004) [14]. They convert the eigenmodes problem to an eigenpairs problem of a finite dimensional matrix. We formulate an approach based on using finite order interpolating polynomials as eigenfunctions for eigenmodes problem. We prove that, under some simple conditions on the RBFs, two approaches converge when increasingly flat BRFs are being used. These results are supported by numerical examples.

For more than one decade, SmC*(d4), SmC*(d3), andSmC*(A) were the only three confirmed commensurate SmC* variant phases with periodicities less than or equal four layers. In 2006, employing ellipsometry and resonant X-ray diffraction (RXRD), our research team first discovered a new liquid crystal mesophase having a six-layer periodicity in one ternary mixture which includes one sulfur-containing compound. From our ellipsometric results, this phase showed antiferroelectric-like optical response. This novel discovery inspired renewed interest to search for liquid crystal mesophases with commensurate periodicities greater than four layers. Soon after, another mesophase having a six-layer structure and showing a ferrielectric-like dielectric response, instead, was uncovered by RXRD measurements on a different binary mixture which has one bromine-containing compound. Meanwhile mesophases having a 5-, 8-, 12- or 15-layer periodicity were reported. However, numerous questions remain to be addressed associated with these unusual reported phases. Theoretical models giving rise to mesophases with periodicities greater than four layers have been developed; but, to date, none of them have provided satisfactory explanations of all the physical phenomena related to the mesophases exhibiting a six-layer structure. Moreover, the question "what is the source of long-range interactions between liquid-like smectic layers, which are responsible for establishing mesophases with long periodicities and mean-field behavior of the smectic-A-smectic-C transition?" remains unanswered for more than three decades.

In 2006, employing ellipsometry and resonant x-ray diffraction, our research group discovered a liquid crystal mesophase having a six-layer periodicity in a ternary mixture (mixture A) as well as in a binary mixture (mixture B). This phase shows antiferroelectric-like properties. Subsequently, J. K. Vij's group used field-induced birefringence to explore the physical properties of various binary mixtures similar to mixture B. Recently, Y. Takanishi et al. obtained dielectric responses and two-dimensional microbeam resonant x-ray diffraction profiles as a function of temperature from a different binary mixture with one compound of the mixture containing a central bromine atom. They discovered another new mesophase which shows a six-layer structure and displays ferrielectric-like responses along with a different phase sequence. This article will review the sequence of events leading up to the discovery of the new phases with a six-layer periodicity and highlight differences in conclusions about the new phases and an ongoing debate about the existence of a phase with five-layer periodicity.

A general approach is given to extend WENO reconstructions to a class of numerical schemes that use different types of moments (i.e., multi-moments) simultaneously as the computational variables, such as point values and grid cell averages. The key is to re-map the multi-moment values to single moment values (e.g., cell average or point values), which can then be used to invoke known, standard reconstruction coefficients and smoothness indicators for single moment WENO reconstructions. The WENO reconstructions in turn provide the numerical approximations for the flux functions and other required quantities. One major advantage of using multi-moments for WENO reconstructions is its compactness. We present two new multi-moment WENO (MM-WENO) schemes of fifth order that use reconstructions supported over only three grid cells, as opposed to the usual five. This is similar to the Hermite WENO schemes of Qiu and Shu (J Comput Phys 193:115–135, 2003 ), which can also be derived using our general approach. Numerical tests demonstrate that the new schemes achieve their designed fifth order accuracy and eliminate spurious oscillations effectively. The numerical solutions to all benchmark tests are of good quality and comparable to the classic, single moment WENO scheme of the same order of accuracy. The basic idea presented in this paper is universal, which makes the WENO reconstruction an easy-to-follow method for developing a wide variety of additional multi-moment numerical schemes.

We will present some recent work on phase-field models for twophase fluids with variable densities. The Quasi-Incompressible Navier-Stokes-Cahn-Hilliard model with the gravitational force being incorporated in the thermodynamically consistent framework will be investigated. Under a minor reformulation of the system we show that there is a continuous energy law underlying the system. For the reformulated system we then design a continuous finite element method and a special temporal scheme such that the energy law is accurately preserved at the discrete level. Such a discrete energy law for a variable density two-phase flow model has never been established before with continuous finite element. We will also show an example that an energy law preserving method will perform better for multiphase flow problems. Finally, we will develop a model to account for the thermocapillary effects. It allows for the different properties (densities, viscosities and heat conductivities) of each component while maintaining thermodynamic consistency. To our knowledge such a model is new. Numerical validation is provided too. The talk is based on two recent papers: [1] Zhenlin Guo and P Lin, A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects, Preprint arXiv: 1401.5793, Jan 2014. http://arxiv.org/pdf/1401.5793v3.pdf [2] Zhenlin Guo, P Lin and JS Lowengrub, A numerical method for the quasi-incompressible CahnHilliard-Navier-Stokes equations for variable density flows with a discrete energy law, J. Comput. Phys. 276 (2014), pp. 486-507. http://arxiv.org/pdf/1402.1402v2.pdf

A WENO re-averaging (or re-mapping) technique is developed that converts function averages on one grid to another grid to high order. Nonlinear weighting gives the essentially non-oscillatory property to the re-averaged function values. The new reconstruction grid is used to obtain a standard high order WENO reconstruction of the function averages at a select point. By choosing the reconstruction grid to include the point of interest, a high order function value can be reconstructed using only positive linear weights. The re-averaging technique is applied to define two variants of a classic CWENO3 scheme that combines two linear polynomials to obtain formal third order accuracy. Such a scheme cannot otherwise be defined, due to the nonexistence of linear weights for third order reconstruction at the center of a grid element. The new scheme uses a compact stencil of three solution averages, and only positive linear weights are used. The scheme extends easily to problems in higher space dimensions, essentially as a tensor product of the one-dimensional scheme. The scheme maintains formal third order accuracy in higher dimensions. Numerical results show that this CWENO3 scheme is third order accurate for smooth problems and gives good results for non-smooth problems, including those with shocks.

With the discovery of the smectic-C(d6)(*) (SmC(d6)(*)) phase showing six-layer periodicity [S. Wang et al., Phys. Rev. Lett. 104, 027801 (2010)] and a recent report of the observation of a possible alternative structure, the need for a reliable and accurate method for distinguishing different possible structures is more urgent than ever. Through simulations using the tensorial structure factor method, we present the resonant x-ray diffraction (RXRD) spectra for different possible structures as proposed in several theoretical studies. Subtle distinctions between models are shown. The ability and limitations of RXRD as a technique for determining the structure of this particular phase is discussed.