Flux Reconstruction Research Articles

Quartz grain microtextures illuminate Pliocene periglacial sand fluxes on the Antarctic continental margin

AbstractOn high‐latitude continental margins sediment is supplied from land to the deep sea through a variety of processes, including iceberg and sea‐ice rafting, and bottom current transport. The accurate reconstruction of sediment fluxes from these sources through time is important in palaeoclimate reconstructions. The goal of this study was to assess a shift in the intensity of glacial processes, iceberg and sea‐ice rafting during the Pliocene through an investigation of coarse sediment deposited at the AND‐2A site in the Ross Sea and at International Ocean Discovery Program Site U1359 on the Antarctic continental rise. Terrigenous particle‐size distributions and suites of quartz grain microtextures in the sand fraction of the deep‐sea sediments were compared to those from Antarctic glaciomarine diamictites as a baseline for proximal glacial sediment in its source area. Using images acquired through Scanning Electron Microscopy, and following a quantitative approach, fewer immature and potentially glacially transported grains were found in Pliocene deep‐sea sand fractions than in ice‐contact sediments. Specifically, in the lower Pliocene interval silt and fine sand percentages are elevated, and microtextures in at least half of the sand fraction are inconsistent with a primary glacial origin. Larger numbers of chemically altered and abraded grains in the deep‐sea sand fraction, along with microtextures that are diagnostic of periglacial environments, suggest a role for eolian sediment transport. These results highlight the anomalous nature of high‐latitude sediment fluxes during prolonged periods of ice retreat. Furthermore, the identification of a significant offshore sediment flux during Antarctic deglaciation has implications for estimated nutrient supply to the Southern Ocean and the potential for high‐latitude climate feedbacks under warmer climate states.

Coupling improved discrete velocity method and G13-based gas kinetic flux solver: A hybrid method and its application for non-equilibrium flows

From the aspect of the multiscale nature of the rarefied flow, a simple hybrid strategy is proposed in this paper in the process of flux reconstruction to couple the improved discrete velocity method (IDVM) and the G13-based gas kinetic flux solver (G13-GKFS). The flow field is divided into the IDVM area and G13 area according to the kinetic nature of the Knudsen layer and the criteria of the local Knudsen number. By eliminating the storage of the distribution functions and evolution of the microscopic equation, the reduction of the computational effort and memory storage can be achieved without sacrificing the accuracy in the whole flow field. Four different non-equilibrium cases from the micro-flow to the supersonic flow are tested by the present hybrid method. The results show good performance and better efficiency. Furthermore, under the framework of the present hybrid method, different non-equilibrium distribution functions with higher-order moments could be employed and coupled easily.

Implementation of a parallel high-order WENO-type Euler equation solver using a CUDA PTX paradigm

ABSTRACT This study proposes the optimization of a low-level assembly code to reconstruct the flux for a splitting flux Harten–Lax–van Leer (SHLL) scheme on high-end graphic processing units. The proposed solver is implemented using the weighted essentially non-oscillatory reconstruction method to simulate compressible gas flows that are derived using an unsteady Euler equation. Instructions in the low-level assembly code, i.e. parallel thread execution and instruction set architecture in compute unified device architecture (CUDA), are used to optimize the CUDA kernel for the flux reconstruction method. The flux reconstruction method is a fifth-order one that is used to process the high-resolution intercell flux for achieving a highly localized scheme, such as the high-order implementation of SHLL scheme. Many benchmarking test cases including shock-tube and four-shock problems are demonstrated and compared. The results show that the reconstruction method is computationally very intensive and can achieve excellent performance up to 5183 GFLOP/s, ∼66% of peak performance of NVIDIA V100, using the low-level CUDA assembly code. The computational efficiency is twice the value as compared with the previous studies. The CUDA assembly code reduces 26.7% calculation and increases 37.5% bandwidth. The results show that the optimal kernel reaches up to 990 GB/s for the bandwidth. The overall efficiency of bandwidth and computation performance achieves 127% of the predicted performance based on the HBM2-memory roofline model estimated by Empirical Roofline Tool.

Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

AbstractIn this article, we clarify reconstruction‐based discretization schemes for unstructured grids and discuss their economically high‐order versions, which can achieve high‐order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux‐and‐solution‐reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended ‐scheme combined with economical flux reconstruction formulas, achieving up to fifth‐order accuracy (sixth‐order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.

A flux reconstruction method for the Korteweg-de Vries equation

In this paper we aim to develop a flux reconstruction (FR) method for the KdV equation, which contains a third derivative term. By introducing two auxiliary variables, the third order derivative is written as a combination of first derivatives, such that the FR schemes for first derivative can be applied directly. To demonstrate the effectiveness of the schemes, numerical results of some benchmark examples are presented, including the examples of a single soliton, double solitons, multiple solitons and an extremely convection-dominated case.

IXV post-flight reconstruction and analysis of the aerothermodynamic measurements along the rarefied portion of the reentry trajectory

Following the reentry of the Intermediate eXperimental Vehicle in 2015, a lot of efforts were carried out at the ONERA for the post-flight analysis and the reconstruction of the heat flux applied to the vehicle. This flight represents a unique means of validation of the computational tools employed for the design of such mission. The aim of this paper is to exploit for the first time these data in the rarefied portion of the vehicle’s reentry. In this context, DSMC simulations were performed between 115 and 90 km and a CFD computation was made for the 90 km flight point. In the standard numerical conditions, different degrees of agreement are obtained and a sensitivity analysis to various numerical and atmospheric parameters provides several ways to significantly improve the results. In addition, the DSMC and CFD comparison provides a good agreement for the surface quantities and suggestions to reach a better level of consistency between both methods and the flight data are finally discussed.

Novel Adaptive Hybrid Discontinuous Galerkin Algorithms for Elliptic Problems

AbstractIn this work, we develop novel adaptive hybrid discontinuous Galerkin algorithms for second-order elliptic problems. For this, two types of reliable and efficient, modulo a data-oscillation term, and fully computable a posteriori error estimators are developed: the first one is a simple residual type error estimator, and the second is a flux reconstruction based error estimator of a guaranteed type for polynomial approximations of any degree by using a simple postprocessing. These estimators can achieve high-order accuracy for both smooth and nonsmooth problems even with high-order approximations. In order to enhance the performance of adaptive algorithms, we introduce𝐾-means clusteringbased marking strategy. The choice of marking parameter is crucial in the performance of the existing strategy such as maximum and bulk criteria; however, the optimal choice is not known. The new strategy has no unknown parameter. Several numerical examples are given to illustrate the performance of the new marking strategy along with our estimators.

Changes in patterns of mineral and chemical elements in bottom sediments of Lake Baikal (Russia) as high-resolution records of moisture for the past 31–16 ka BP

The high-resolution biochemical records (SRXRF, ICP-MS and FTIR analyses) of bottom sediments were used for the reconstruction of clastic flux into Lake Baikal via the Barguzin River as moisture and glacier indexes from 1631 ka BP. According to the obtained data, proxies of a high supply of clastic flux by the river are high: the content of feldspar and LREE/HREE, Rb/Sr and Th/Mo ratios. We sampled a sediment core extremely close to the estuary of the Barguzin River. It seems that the water budget of the Baikal tributaries dramatically reduced ca. 34–30 ka BP, and the lake water level was minimum (less than 35–40 m compared to the present) for the last 34 ka. The lake water level has gradually increased from ca. 30 to 25–23 ka BP, and it was likely linked with the activity of the East Asia Monsoon. The very contrasting changes in chemical and mineralogical records were ca. 29.5 ka BP, and these changes were very likely associated with the Heinrich-3; however, there were no significant changes in the records around the Heinrich-2. Our records revealed some intensification of the clastic flux of the Barguzin River into the Barguzin Bay ca. 25–23 ka BP. The Baikal event ca. 21.5–22.5 ka BP was characterized by a sharp reduction of the influx of clastic material into the Barguzin Bay. The time span of 19.5–18 ka BP was the last time that the Barguzin River inflow increased during the Late Pleistocene. This increase in the water budget was at arid regional conditions, and we assumed it could occur due to the contribution of meltwater from glaciers of the Barguzin Ridge. It is most likely that at the regional Late Pleistocene/Holocene transition (around 12–13 ka BP), a rapid rise of the water budget of the Barguzin River occurred, while the lake level was not high, and the sediments deposited from 12(13)–16.5 ka BP could be directly eroded by a kill of the Barguzin River. In general, the obtained records show that an increase in the water budget of the Barguzin River was gradual. The duration of high inflow into the Barguzin Bay was ca. 1.5 ka, while the reductions of this inflow were dramatic.

East Asian monsoon intensification promoted weathering of the magnesium-rich southern China upper crust and its global significance

The Oligocene-Miocene boundary Asian climatic reorganization linked to the northward migration of the East Asian monsoon into subtropical China is a potentially important but poorly constrained atmospheric CO2 consumption process. Here, we performed a first-order estimate of the CO2 consumption induced by silicate chemical weathering and organic carbon burial in subtropical China related to this climatic reorganization. Our results show that an increase in long-term CO2 consumption by silicate weathering varies from 0.06×1012 to 0.87×1012 mol yr−1 depending on erosion flux reconstructions, with an ~50% contribution of Mg-silicate weathering since the late Oligocene. The organic carbon burial flux is approximately 25% of the contemporary CO2 consumption by silicate weathering. The results highlight the significant role of weathering of the Mg-rich upper continental crust in East China, which would contribute to the rapid decline in atmospheric CO2 during the late Oligocene and the Neogene rise in the seawater Mg content. If this climatic reorganization was mainly induced by the Tibetan Plateau uplift, our study suggests that the growth of the Himalayan-Tibetan Plateau can lead to indirect modification of the global carbon and magnesium cycles by changing the regional hydrological cycle in areas of East Asia that are tectonically less active.

AN IMPLICIT BLOCK JACOBI APPROACH FOR HIGH-ORDER FLUX RECONSTRUCTION METHOD

Recently, the flux reconstruction (FR) method has attracted more and more attentions for its simplicity and generality. However, it is still computationally expensive and time consuming when simulating the complex flow problems by FR method. There is a huge demand for developing appropriate efficient implicit solvers and parallel computing techniques for FR. This paper proposes an implicit high-order flux reconstruction solver on GPU platform based on the block Jacobi iteration method. As it is inefficient to solve the large global linear system resulting from spatial and implicit temporal discretization of FR directly. A block Jacobi approach is used to change the characteristics of the lift-hand matrix of the global linear system and this avoids the dependence of neighboring elements. Therefore, only the diagonal blocks of global matrix need to be stored and calculated. Then, the problem of solving the huge global linear system is transformed into solving a series of local linear equations simultaneously. Finally, these small local linear equations would be solved by the LU decomposition method in parallel on GPU platforms. Two typical cases, including subsonic flows over a bump and a NACA0012 airfoil, were simulated and compared with the multi-grid explicit Runge-Kutta scheme. The numerical results demonstrated that the present implicit method can reduce the iterations significantly. Meanwhile, the implicit solver has shown at least 10x speedup over the multi-grid Runge-Kutta scheme in all cases.

On False Accuracy Verification of UMUSCL Scheme

In this paper, we reveal a mechanism behind a false accuracy verification encountered with unstructured-grid schemes based on solution reconstruction such as UMUSCL. Third- (or higher-) order of accuracy has been reported for the Euler equations in the literature, but UMUSCL is actually second-order accurate at best for nonlinear equations. False high-order convergence occurs generally for a scheme that is high order for linear equations but second-order for nonlinear equations. It is caused by unexpected linearization of a target nonlinear equation due to too small of a perturbation added to an exact solution used for accuracy verification. To clarify the mechanism, we begin with a proof that the UMUSCL scheme is third-order accurate only for linear equations. Then, we derive a condition under which the third-order truncation error dominates the second-order error and demonstrate it numerically for Burgers' equation. Similar results are shown for the Euler equations, which disprove some accuracy verification results in the literature. To be genuinely third-order, UMUSCL must be implemented with flux reconstruction.

Intensified organic carbon burial on the Australian shelf after the Middle Pleistocene transition

The Middle Pleistocene Transition (MPT) represents a major change in Earth's climate state, exemplified by the switch from obliquity-dominated to ∼100-kyr glacial/interglacial cycles. To date, the causes of this significant change in Earth's climatic response to orbital forcing are not fully understood. Nonetheless, this transition represents an intrinsic shift in Earth's response to orbital forcing, without fundamental changes in the astronomical rhythms. This study presents new high-resolution records of International Ocean Discovery Program (IODP) Site U1460 (eastern Indian Ocean, 27°S), including shallow marine productivity and organic matter flux reconstructions. The proxy series covers the interval between 1.1 and 0.6 Ma and provides insights into Pleistocene Leeuwin Current dynamics along the West Australian shelf. The large >45 m global sea level drop during the marine isotope stage (MIS) 22–24 is marked in our data, suggesting that the MPT led to large-scale changes in Indian Ocean circulation patterns and surface water conditions. We consider shelf exposure (and thus the “Sahul-Indian Ocean Bjerknes mechanism”) as a possible key process to increase the upwelling of nutrient-rich sub-Antarctic Mode waters through the Leeuwin Undercurrent along the Australian shelf. We conclude that the shoaling of nutrient-rich lower-thermocline waters enhanced mid-latitude productivity patterns in the eastern Indian Ocean across the 900-ka event.

Finite-volume flux reconstruction and semi-analytical particle tracking on triangular prisms for finite-element-type models of variably-saturated flow

Consistent particle tracking relies on conforming velocity fields that ensure local mass conservation on elements. Cell-centered finite-volume and mixed finite-element methods result in conforming velocity fields but this is not the case for continuous Galerkin methods, such as the standard finite element method (FEM). Nonetheless standard FEM is often used for subsurface flow modeling because it yields a continuous approximation of hydraulic heads, and it naturally handles unstructured grids and full material tensors. Acknowledging these advantages and the wide-spread use of finite-element-type simulations, we present a postprocessing method that reconstructs a cell-centered finite-volume solution from a finite-element-type solution of the variably-saturated subsurface flow equation to obtain conforming, mass-conservative fluxes. Using the linear average velocity field derived from these fluid fluxes, we employ element-wise analytical solutions for triangular prisms to compute particle trajectories and associated travel times. As a result, we can compute consistent particle trajectories for variably-saturated flow solutions generated by node-centered methods, such as finite element or finite difference methods, that do not yield conforming velocity fields. Our flux reconstruction solves a linear elliptic problem whose size is on the order of the number of elements, which is computationally much faster than solving the initial, non-linear transient variably-saturated flow equation. Compared to other postprocessing schemes, our flux reconstruction is numerically stable, fast to compute, and does not induce severe numerical artifacts when applied to heterogeneous domains with strongly varying velocities. However, these advantages come with a comparably high coding effort and the necessity of solving a global system of equations. We show that the results of our flux reconstruction are close to the node-centered primal solution for variably saturated three-dimensional flow with heterogeneous material properties.

High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.

A high order flux reconstruction interface capturing method with a phase field preconditioning procedure

This paper presents a simple and highly accurate method for capturing sharp interfaces moving in divergence-free velocity fields using the high-order Flux Reconstruction approach on unstructured grids. A well-known limitation of high-order methods is their susceptibility to the Gibbs phenomenon; the appearance of spurious oscillations in the vicinity of discontinuities and steep gradients makes it difficult to accurately resolve shocks or sharp interfaces. In order to address this issue in the context of sharp interface capturing, a novel, preconditioned and localized phase-field method is developed in this work. The numerical accuracy of interface normal vectors is improved by utilizing a preconditioning procedure based on the level set method with localized artificial viscosity stabilization. The developed method was implemented in the framework of the multi-platform Flux Reconstruction open-source code PyFR [1]. Kinematic numerical tests in 2D and 3D conducted on different mesh types showed that the preconditioning procedure significantly improves accuracy with little added computational effort. The results demonstrate the conservativeness of the proposed method and its ability to capture highly distorted interfaces with superior accuracy when compared to conventional and high-order VOF and level set methods. Dynamic interface capturing validation tests were also carried out by coupling the proposed method to the entropically damped artificial compressibility variant of the incompressible Navier–Stokes equations. The high accuracy and locality of the proposed method offer a promising route to carrying out massively-parallel, high accuracy simulations of multi-phase, incompressible phenomena.

High-order implicit large-eddy simulation of flow over a marine propeller

We report the first high-order eddy-resolving simulation of flow over a marine propeller using a recently developed high-order sliding-mesh method. This method employs the flux reconstruction framework and a new dynamic curved mortar approach to handle the complex rotating geometries. For a wide range of working conditions, it is validated to predict the loads very accurately against experiments. The method’s low-dissipation characteristic has allowed the capturing of a broad spectrum of turbulence structures for very long distances even on a very coarse grid. Comparison with a previous low-order simulation is also carried out to show the low-dissipation advantage of the present simulations. From detailed load analysis, the major loads and their distributions and time and frequency scales are identified. Visualizations of the instantaneous, phase-averaged, and time-averaged flow fields have revealed the processes of tip vortex formation, major vortex evolutions, and flow instability developments at different working conditions. The effects of different fairwaters on the propeller’s overall performance are also quantitatively assessed.

A Comparative Study from Spectral Analyses of High-Order Methods with Non-Constant Advection Velocities

The response of numerical methods to flow properties dynamic changes is a key ingredient of numerical modeling calibration. In this context, a range of spectral analyses and canonical fluid mechanics problems are explored to compare the responses of the high-order spectral difference scheme and of the flux reconstruction methods. Spatial eigen-analysis (Hu et al. in J Comput Phys 151(2):921–946, 1999; Hu and Atkins in J Comput Phys 182(2):516–545, 2002), based on spatially evolving oscillations, is used to get useful insights for problems with inflow/outflow boundary conditions (i.e., non-periodic), while non-modal analysis (Fernandez et al. in Comput Methods Appl Mech Eng 346:43–62, 2019. https://doi.org/10.1016/j.cma.2018.11.027 ) focusses on the short-term dynamics of the discretised system. These two approaches are also extended to the general scalar conservation law with non-constant advection velocity. All these tools are used to obtain more insight about the expected behavior of the mentioned numerical methods for typical engineering applications. On this regard, despite the lack of a mathematical proof of stability in the non-linear case, the selected numerical schemes, for which an extensive literature is available for applications to Euler and Navier–Stokes equations, will be assumed to be stable. Findings are then verified considering one-dimensional linear advection, and two- and three-dimensional flows modeled with the compressible Euler equations. Implications in the accurate control of numerical dissipation for turbulent flows is also addressed. Within the context of high-order methods, this work complement the former temporal spectral analyses of the discontinuous Galerkin approach discretising the linear advection equation.

Simulation of flow induced by single-dielectric-barrier-discharge plasma actuator using a high-order flux-reconstruction scheme

The starting vortex flow generated by a single-dielectric-barrier-discharge (SDBD) plasma actuator in still air is investigated numerically by direct numerical simulation (DNS) using a 4th order Navier–Stokes solver and Suzen's model for the plasma actuator. The Navier–Stokes equations and the equations for the electric field and the charge density distribution in the plasma body force model are solved by the same high-order method based on the flux reconstruction algorithm on arbitrary mixed grids, allowing easy extension to configurations of complex geometry. The computational method and its high orders of accuracy are verified for the problem of the translation of an isentropic vortex and validated for the vortex shedding problem behind a circular cylinder. The plasma body force model is then implemented in the Navier–Stokes code to perform highly resolved DNS of the starting vortex generated by an SDBD plasma actuator. The computed vortex pattern and its trajectory with time are compared with available experimental data. The computational results confirm similarity laws on the motion of the vortex.

A data-driven physics-informed finite-volume scheme for nonclassical undercompressive shocks

We propose a data-driven physics-informed finite-volume scheme for the approximation of small-scale dependent shocks. Nonlinear hyperbolic conservation laws with non-convex fluxes allow nonclassical shock wave solutions. In this work, we consider the cubic scalar conservation law as representative of such systems. As standard numerical schemes fail to approximate nonclassical shocks, schemes with controlled dissipation and schemes with well-controlled dissipation have been introduced by LeFloch and Mohammadian and by Ernest and coworkers, respectively. Emphasis has been placed on matching the truncation error of the numerical scheme with physically relevant small-scale mechanisms. However, aforementioned schemes can introduce oscillations as well as excessive dissipation around shocks. In our approach, a convolutional neural network is used for an adaptive nonlinear flux reconstruction. Based on the local flow field, the network combines local interpolation polynomials with a regularization term to form the numerical flux. This allows to modify the discretization error by nonlinear terms. In a supervised learning task, the model is trained to predict the time evolution of exact solutions to Riemann problems. The model is physics-informed as it respects the underlying conservation law. Numerical experiments for the cubic scalar conservation law show that the resulting method is able to approximate nonclassical shocks very well. The adaptive reconstruction suppresses oscillations and enables sharp shock capturing. Generalization to unseen shock configurations, smooth initial value problems, and shock interactions is robust and shows very good results.

Accuracy, stability, and performance comparison between the spectral difference and flux reconstruction schemes

We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation upon which to provide a fair and accurate assessment of these two schemes in terms of accuracy, stability, and performance with special attention to the true spectral difference scheme and the modified spectral difference scheme recovered via the flux reconstruction formulation. Building on previous analysis of the spectral difference and flux reconstruction schemes, we provide a novel nonlinear stability analysis of the spectral difference scheme. Through various numerical experiments, we demonstrate the additional stability afforded by the true, baseline spectral difference scheme without explicit filtering or de-aliasing due to its inherent feature of staggered flux points. This arrangement leads to favorable suppression of aliasing errors and improves stability needed for under-resolved simulations of turbulent flows.