Least-squares Reconstruction Research Articles

Implementation of a high-order spatial discretization into a finite volume solver: Applications to turbomachinery test cases using an eddy-viscosity turbulence closure

In this study, the implementation of a high-order spatial discretization method into a Finite Volume solver is presented. Specific emphasis is put on the analysis of the performance over selected turbomachinery test cases. High-order numerical discretization is achieved by adopting the cell-centered Least-Square reconstruction, which is implemented in the in-house solver HybFlow. The validation of the adopted methodology is performed by assessing the solution of a turbulent flat plate with zero pressure gradient, using a eddy-viscosity transitional model. The test case also evidences the effect of the discretization of gradient-based source terms when a high-order reconstruction methodology is used. In the second part of the paper, the solver is used for the solution of relevant two-dimensional turbomachinery test cases, assessing the impact of 2nd and 3rd order reconstruction on the prediction of the aerodynamics and the heat transfer for respectively a low-pressure blade and a high-pressure turbine vane. It is shown how a high-order reconstruction allows for obtaining a better prediction of turbomachinery aerodynamics, with lower number of elements. The benefits over heat transfer predictions in high Reynolds number conditions are instead limited to the reduction of heat transfer coefficient spikes in under-resolved regions of the blade. Eventually, the methodology is validated for a three-dimensional low-pressure turbine cascade with realistic boundary layer inflow conditions.

A high-order finite difference method for moving immersed domain boundaries and material interfaces

We present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

Non-ideal magnetohydrodynamics on a moving mesh II: Hall effect

ABSTRACT In this work, we extend the non-ideal magnetohydrodynamics (MHD) solver in the moving-mesh code arepo to include the Hall effect. The core of our algorithm is based on an estimation of the magnetic field gradients by a least-squares reconstruction on the unstructured mesh, which we also used in the companion paper for Ohmic and ambipolar diffusion. In an extensive study of simulations of a magnetic shock, we show that without additional magnetic diffusion our algorithm for the Hall effect becomes unstable at high resolution. We can however stabilize it by artificially increasing the Ohmic resistivity, ηOR, so that it satisfies the condition ηOR ≥ ηH/5, where ηH is the Hall diffusion coefficient. Adopting this solution, we find second-order convergence for the C-shock and are also able to accurately reproduce the dispersion relation of the whistler waves. As a first application of the new scheme, we simulate the collapse of a magnetized cloud with constant Hall parameter ηH and show that, depending on the sign of ηH, the magnetic braking can either be weakened or strengthened by the Hall effect. The quasi-Lagrangian nature of the moving-mesh method used here automatically increases the resolution in the forming core, making it well suited for more realistic studies with non-constant magnetic diffusivities in the future.

Two-step multi-resolution reconstruction-based compact gas-kinetic scheme on tetrahedral mesh

This paper presents the development of a third-order compact gas-kinetic scheme (GKS) for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme utilizes a time-dependent gas distribution function at a cell interface to not only calculate the fluxes needed for updating the cell-averaged flow variables but also to evaluate the flow variables at the cell interface. This leads to the evolution of cell-averaged gradients of flow variables. The success of this scheme heavily relies on the initial data reconstruction techniques, with an emphasis on their application to the tetrahedral mesh. Employing a conventional second-order unlimited least-square reconstruction directly on the cell-averaged flow variables of von Neumann neighbouring cells can introduce linear instability into the scheme. However, by using the updated cell-averaged gradients, the GKS with a third-order compact smooth reconstruction remains linearly stable under a large CFL number when applied to a tetrahedral mesh. To enhance the robustness of the high-order compact GKS for capturing a discontinuous solution, we propose a novel two-step multi-resolution weighted essentially non-oscillatory (WENO) reconstruction. This innovative approach overcomes the stability issues associated with a second-order compact reconstruction by incorporating a pre-reconstruction step. Additionally, it simplifies the third-order non-linear reconstruction process by adding a single large stencil to those used in the second-order one. A high-order wall boundary condition is achieved by fusing the constrained least-square technique with the WENO procedure, where a quadratic element is used in the reconstruction for cells with a curved boundary. Numerical tests involving both the second-order and third-order compact GKS are presented, encompassing both inviscid and viscous flows at both low and high speeds. The results demonstrate that the proposed third-order compact scheme possesses robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.

Constructive subsampling of finite frames with applications in optimal function recovery

In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in Cm. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds 0<A≤B<∞ (and condition B/A) a similarly conditioned reweighted subframe consisting of merely O(mlog⁡m) elements. Further, utilizing a deterministic subsampling method based on principles developed by Batson, Spielman, and Srivastava to control the spectrum of sums of Hermitian rank-1 matrices, we are able to reduce the number of elements to O(m) (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This permits the derivation of new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for L2(D,ν) with constructible node sets of size O(m) for m-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.

Learning Stable Models for Prediction and Control

In this article, we demonstrate the benefits of imposing stability on data-driven Koopman operators. The data-driven identification of stable Koopman operators (DISKO) is implemented using an algorithm [1] that computes the nearest <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stable</i> matrix solution to a least-squares reconstruction error. As a first result, we derive a formula that describes the prediction error of Koopman representations for an arbitrary number of time steps, and which shows that stability constraints can improve the predictive accuracy over long horizons. As a second result, we determine formal conditions on basis functions of Koopman operators needed to satisfy the stability properties of an underlying nonlinear system. As a third result, we derive formal conditions for constructing Lyapunov functions for nonlinear systems out of stable data-driven Koopman operators, which we use to verify stabilizing control from data. Finally, we demonstrate the benefits of DISKO in prediction and control with simulations using a pendulum and a quadrotor and experiments with a pusher-slider system. The paper is complemented with a video: <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://sites.google.com/view/learning-stable-koopman</uri> .

Modeling the Solar Corona with an Implicit High-order Reconstructed Discontinuous Galerkin Scheme

The present study aims to apply an implicit high-order reconstructed discontinuous Galerkin (DG) scheme (rDG(P 1 P 2)) to simulate the steady-state solar corona. In this scheme, a piecewise quadratic polynomial solution, P 2, is obtained from the underlying piecewise linear DG solution, P 1, by least-squares reconstruction with a weighted essentially nonoscillatory limiter. The reconstructed quadratic polynomial solution is then used for the computation of the fluxes and source terms. In addition, an implicit time integration method with large time steps is considered in this work. The resulting large linear algebraic system of equations from the implicit discretization is solved by the cellwise relaxation implicit scheme which can make full use of the compactness of the DG scheme. The code of the implicit high-order rDG scheme is developed in Fortran language with message passing interface parallelization in Cartesian coordinates. To validate this code, we first test a problem with an exact solution, which confirms the expected third-order accuracy. Then we simulate the solar corona for Carrington rotations 2167, 2183, and 2210, and compare the modeled results with observations. We find that the numerical results basically reproduce the large-scale observed structures of the solar corona, such as coronal holes, helmet streamers, pseudostreamers, and high- and low-speed streams, which demonstrates the capability of the developed scheme.

Effects of Shock Waves on Shack–Hartmann Wavefront Sensor Data

Shock waves will form by turning supersonic or locally supersonic flow and result in an increase in the freestream density downstream of the shock. This increase leads to optical distortions that limit the effectiveness of aircraft-mounted laser systems. In this paper, analytic expressions are developed to describe these optical distortions in terms of the optical-path difference (OPD). Pupil-plane disturbances imposed by the shock are studied for two cases: when the shock is parallel to the propagation direction and when the shock is on an angle relative to the propagation direction. Upon propagation from the pupil plane, the analysis shows that shock-induced phase discontinuities can sometimes cause the irradiance pattern in the image plane to bifurcate. Despite a large amount of tilt in the pupil plane, the bifurcated irradiance pattern does not map to a proportional shift in the image plane. The implications that these findings have on Shack–Hartmann wavefront sensor (SHWFS) data are also explored. The results show that least-squares reconstruction from the SHWFS data yield accurate estimates of the change in OPD across the shock when the magnitude of the phase difference caused by the shock is between 0 and approximately . However, when , the results show that least-squares reconstruction begins to severely underestimate the change in OPD across the shock. Such results will inform future efforts looking to develop aircraft-mounted laser systems.

Eilmer: An open-source multi-physics hypersonic flow solver

This paper introduces Eilmer, a general-purpose open-source compressible flow solver developed at the University of Queensland, designed to support research calculations in hypersonics and high-speed aerothermodynamics. Eilmer has a broad userbase in several university research groups and a wide range of capabilities, which are documented on the project's website, in the accompanying reference manuals, and in an extensive catalogue of example simulations. The first part of this paper describes the formulation of the code: the equations, physical models, and numerical methods that are used in a basic fluid dynamics simulation, as well as a handful of optional multi-physics models that are commonly added on to do calculations of hypersonic flow. The second section describes the processes used to develop and maintain the code, documenting our adherence to good programming practice and endorsing certain techniques that seem to be particularly helpful for scientific codes. The final section describes a half-dozen example simulations that span the range of Eilmer's capabilities, each consisting of some sample results and a short explanation of the problem being solved, which together will hopefully assist new users in beginning to use Eilmer in their own research projects. Program summaryProgram Title: EilmerCPC Library link to program files:https://doi.org/10.17632/gy2ds2fyxm.1Developer's repository link:https://github.com/gdtk-uq/gdtkCode Ocean capsule:https://codeocean.com/capsule/7226427Licensing provisions: GPLv3Programming language: D, LuaSupplementary material:https://gdtk.uqcloud.netNature of problem: Eilmer solves the compressible Navier-Stokes equations with a particular emphasis on flows at hypersonic speeds. The code includes modelling for high-temperature gas effects such as chemical and vibrational nonequilibrium. Eilmer can be used for the simulation for unsteady and steady flows.Solution method: The code is implemented in D [1] and built on a finite-volume formulation that is capable of solving the Navier-Stokes equations in 2D and 3D computational domains, discretised with structured or unstructured grids. Grids may be generated using a built-in parametric scripting tool or imported from commercial gridding software. The inviscid fluxes are computed using the reconstruction-evolution approach. In structured-grid mode, reconstruction stencils up to fourth-order spatial accuracy are available. In unstructured-grid mode, least-squares reconstruction provides second-order spatial accuracy. A variety of flux calculators are available in the code. Viscous fluxes are computed with compact stencils with second-order spatial accuracy. For unsteady flows, explicit time-stepping with low-order RK-family schemes are available, along with a point-implicit Backward-Euler update scheme for stiff systems of equations. For steady flows, convergence can be greatly accelerated using a Jacobian-free Newton-Krylov update scheme, which seeks a global minimum in the residuals using a series of large pseudo-timesteps. Domain decomposition is used for parallel execution using both shared memory and distributed memory programming techniques.Additional comments including restrictions and unusual features: Eilmer provides a programmable interface for pre-processing, post-processing and user run-time customisations. The programmable interface is enabled using a built-in embedded interpreter for the Lua programming language [2]. Run-time customisations include used-defined boundary conditions, source terms and grid motion.

A Three-Order, Divergence-Free Scheme for the Simulation of Solar Wind

In this paper, we present a three-order, divergence-free finite volume scheme to simulate the steady state solar wind ambient. The divergence-free condition of the magnetic field is preserved by the constrained transport (CT) method. The CT method can keep the magnetic fields divergence free if the magnetic fields is divergence free initially. Thus, a least-squares reconstruction of magnetic field with the divergence free constraints is used to make the magnetic fields global solenoidality initially. High order spatial accuracy is obtained through a non-oscillatory hierarchical reconstruction, while a high order time discretization is achieved using a three-order Runge–Kutta scheme. This new model of three order in space and time is validated by numerical results for Carrington rotation 2207. The numerical results show its capability for producing stable reliable results for structured solar wind. The high-order, divergence-free properties of this method make it an ideal tool for the simulations of coronal mass ejection in future.

A reconstructed discontinuous Galerkin method based on variational formulation for compressible flows

A new reconstructed discontinuous Galerkin (rDG) method based on variational formulation is developed for compressible flows. In the presented method, a higher-order piece-wise polynomial is reconstructed based on the underlying discontinuous Galerkin (DG) solution. This reconstruction is done by using a newly developed variational formulation. The variational reconstruction (VR) can be seen as an extension of the compact finite difference (FD) schemes to unstructured grids. The reconstructed variables are obtained by solving an extreme-value problem, which minimizes the jumps of the reconstructed piece-wise polynomials across the cell interfaces, and therefore maximizes the smoothness of the reconstructed solution. Intrinsically, the stencils of the presented reconstruction are the entire mesh, so this method is robust even on tetrahedral grids. A variety of benchmark test cases are presented to assess the accuracy, efficiency and robustness of this rDG method. The numerical experiments demonstrate that the developed rDG method based on variational formulation can maintain the linear stability, obtain the designed high-order accuracy, and outperform the rDG counterpart based on the least-squares reconstruction for both inviscid and viscous compressible flows.

GVOF: An open-source package for unsplit geometric volume of fluid methods on arbitrary grids

The gVOF package implements several accurate and efficient geometric volume of fluid (VOF) methods on arbitrary grids, either structured or unstructured with convex or non-convex cells, based on multidimensional unsplit advection and piecewise linear interface calculation (PLIC) schemes, with the purpose of facilitating and extending the use of advanced unsplit geometric VOF methods in new or existing computational fluid dynamics codes. The package includes a complete and self-contained set of routines for VOF initialization, interface reconstruction and fluid advection, and uses as external libraries a set of publicly available in-house tools to perform several analytical and geometrical operations. These operations may involve handling of high-complex non-convex flux polyhedra, even with self-intersecting faces, which are robustly and efficiently treated in this work without the need of costly techniques based on convex decomposition. Results for the accuracy, computational efficiency, and volume (local and global) conservation properties of different combinations of the implemented advection and reconstruction methods are presented for several numerical tests on structured and unstructured grids. An extensive comparison with results obtained by other authors using advanced geometric VOF methods shows the outstanding performance of the gVOF package in terms of efficiency and accuracy. To demonstrate the performance of the package in solving complex two-phase flow problems, the implemented methods are combined with an existing in-house code to simulate the impact of a water drop on a free surface. Program summaryProgram Title:gVOFCPC Library link to program files:https://doi.org/10.17632/9gmn6gsb6p.1Licensing provisions: GPLv3Programming language:FORTRAN and C, with C interfacesExternal routines/libraries:VOFTools (https://doi.org/10.17632/brrgt645bh.3) and isoap (https://doi.org/10.17632/4rcf98s74c.1) librariesNature of problem: The software package includes efficient and accurate routines for volume of fluid initialization, reconstruction of interfaces and fluid advection on arbitrary grids, either structured or unstructured with convex or non-convex cells, which are used to implement advanced unsplit geometric VOF methods. In particular, the package includes a fluid volume fraction initialization method, six PLIC reconstruction methods and three multidimensional unsplit advection methods. Routines to visualize interfaces and compute reconstruction errors; tests to assess the accuracy, computational efficiency, and volume conservation of the implemented methods for the reconstruction and advection of complex interfaces on arbitrary grids; and a user manual have also been included in the supplied package.Solution method: Basically, the implemented methods, which can be used on grids with polyhedral cells of arbitrary geometry, have the following general characteristics. The methods implemented for fluid volume fraction initialization and reconstruction error calculation are based on a recursive grid refinement procedure to compute the fluid volume bounded by cell edges and an implicitly-defined fluid interface. The six implemented interface reconstruction methods are the following: a least-squares gradient interface reconstruction method; three isosurface-based interface reconstruction methods; an extension to 3D of the iterative Swartz interface reconstruction; and a least-squares fit interface reconstruction method. Three unsplit advection methods, which are based on edge-matched, face-matched and non-matched flux polyhedra and are valid for 3D arbitrary grids, are implemented. All the implemented methods can be combined by the user to solve different tests on arbitrary grids. The implemented routines can be used in FORTRAN or C. The OpenMP application programming interface is used to improve the computational efficiency.

A vertex-centered finite volume method with interface sharpening technique for compressible two-phase flows

A robust and efficient finite volume method with interface sharpening technique has been developed to solve the six-equation multi-fluid single-pressure model for compressible two-phase flows. The numerical method is implemented in a three-dimensional vertex-centered code. A least-squares reconstruction with Kuzmin's vertex-based (VB) limiter is implemented for the volume fraction and a set of primitive variables in the presented finite volume framework. In regions where two different fluid components are present within a cell, a sharpening technique based on THINC (Tangent of Hyperbola for Interface Capturing) is adopted to provide a sharp resolution for the transitioning interface. These reconstructed values are then used as the initial data for Riemann problems. The enhanced AUSM+ -up scheme is applied to both liquid and gas flows. The multi-stage Runge-Kutta method is used for time marching. A number of benchmark test cases are presented to assess the performance of the present method. These include: an air-water interface moving at a constant velocity, Ransom's faucet problem, air-water/water-air shock tube problems with high pressure ratios, a shock in air impacting a water column case, an underwater explosion case and an air bubble blast case. In all of these cases, the shock and rarefaction waves are captured accurately, especially with the THINC interface sharpening technique.

A Provably Positive, Divergence-free Constrained Transport Scheme for the Simulation of Solar Wind

Abstract In this paper, we present a provably positive, divergence-free constrained transport (CT) scheme to simulate the steady-state solar wind ambient with the three-dimensional magnetohydrodynamics numerical model. The positivity can be lost in two ways: one way is in the reconstruction process, and the other is in the updating process when the variables are advanced to the next time step. We adopt a self-adjusting strategy to bring the density and pressure into the permitted range in the reconstruction process, and use modified wave speeds in the Harten–Lax–van Leer flux to ensure the positivity in the updating process. The CT method can keep the magnetic fields divergence-free if the magnetic fields are divergence-free initially. Thus, we combine the least-squares reconstruction of the magnetic fields with the divergence-free constraints to make the magnetic fields globally solenoidal initially. Furthermore, we adopt a radial basis function method to interpolate variables at boundaries that can keep the magnetic field locally divergence-free. To verify the capability of the model in producing structured solar wind, the modeled results are compared with Parker Solar Probe (PSP) in situ observations during its first two encounters, as well as Wind observations at 1 au. Additionally, a solar maximum solar wind background is simulated to show the property of the model’s ability to preserve the positivity. The results show that the model can provide a relatively satisfactory comparison with PSP or Wind observations, and the divergence error is about 10−10 for all of the tests in this paper.

A polarization tensor approximation for the Hessian in iterative solvers for non-linear inverse problems

For many inverse parameter problems for partial differential equations in which the domain contains only well-separated objects, an asymptotic solution to the forward problem involving ‘polarization tensors’ exists. These are functions of the size and material contrast of inclusions, thereby describing the saturation component of the non-linearity. In this paper, we show how such an asymptotic series can be applied to non-linear least-squares reconstruction problems, by deriving an approximate diagonal Hessian matrix for the data misfit term. Often, the Hessian matrix can play a vital role in dealing with the non-linearity, generating good update directions which accelerate the solution towards a global minimum, but the computational cost can make direct calculation infeasible. Since the polarization tensor approximation assumes sufficient separation between inclusions, our approximate Hessian does not account for non-linearity in the form of lack of superposition in the inverse problem. It does, however, account for the non-linear saturation of the change in the data with increasing material contrast. We, therefore, propose to use it as an initial Hessian for quasi-Newton schemes. We present numerical experimentation into the accuracy and reconstruction performance of the approximate Hessian for the case of electrical impedance tomography, providing a proof of principle of the reconstruction scheme.

Cycling Rate-Induced Spatially-Resolved Heterogeneities in Commercial Cylindrical Li-Ion Batteries.

Synchrotron high-energy X-ray diffraction computed tomography has been employed to investigate, for the first time, commercial cylindrical Li-ion batteries electrochemically cycled over the two cycling rates of C/2and C/20. This technique yields maps of the crystalline components and chemical species as a cross-section of the cell with high spatiotemporal resolution (550 × 550 images with 20 × 20 × 3 µm3 voxel size in ca. 1 h). The recently developed Direct Least-Squares Reconstruction algorithm is used to overcome the well-known parallax problem and led to accurate lattice parameter maps for the device cathode. Chemical heterogeneities are revealed at both electrodes and are attributed to uneven Li and current distributions in the cells. It is shown that this technique has the potential to become an invaluable diagnostic tool for real-world commercial batteries and for their characterization under operating conditions, leading to unique insights into "real" battery degradation mechanisms as they occur.

A Photogrammetric-Photometric Stereo Method for High-Resolution Lunar Topographic Mapping Using Yutu-2 Rover Images

Topographic products are important for mission operations and scientific research in lunar exploration. In a lunar rover mission, high-resolution digital elevation models are typically generated at waypoints by photogrammetry methods based on rover stereo images acquired by stereo cameras. In case stereo images are not available, the stereo-photogrammetric method will not be applicable. Alternatively, photometric stereo method can recover topographic information with pixel-level resolution from three or more images, which are acquired by one camera under the same viewing geometry with different illumination conditions. In this research, we extend the concept of photometric stereo to photogrammetric-photometric stereo by incorporating collinearity equations into imaging irradiance model. The proposed photogrammetric-photometric stereo algorithm for surface construction involves three steps. First, the terrain normal vector in object space is derived from collinearity equations, and image irradiance equation for close-range topographic mapping is determined. Second, based on image irradiance equations of multiple images, the height gradients in image space can be solved. Finally, the height map is reconstructed through global least-squares surface reconstruction with spectral regularization. Experiments were carried out using simulated lunar rover images and actual lunar rover images acquired by Yutu-2 rover of Chang’e-4 mission. The results indicate that the proposed method achieves high-resolution and high-precision surface reconstruction, and outperforms the traditional photometric stereo methods. The proposed method is valuable for ground-based lunar surface reconstruction and can be applicable to surface reconstruction of Earth and other planets.

A third-order subcell finite volume gas-kinetic scheme for the Euler and Navier-Stokes equations on triangular meshes

A third-order gas-kinetic scheme (GKS) based on the subcell finite volume (SCFV) method is developed for the Euler and Navier-Stokes equations on triangular meshes, in which a computational cell is subdivided into four subcells. The scheme combines the compact high-order reconstruction of the SCFV method with the high-order flux evolution of the gas-kinetic solver. Different from the original SCFV method using weighted least-square reconstruction along with a conservation correction, a constrained least-square reconstruction is adopted in SCFV-GKS so that the correction is not necessary and continuous polynomials inside each main cell can be reconstructed. For flows with large gradients, a compact hierarchical WENO limiting is adopted. In the gas-kinetic flux solver, the inviscid and viscous flux are coupled and computed simultaneously. Moreover, the spatial and temporal evolution are coupled nonlinearly which enables the developed scheme to achieve third-order accuracy in both space and time within a single stage. As a result, no quadrature point is required for the flux evaluation and the multi-stage Runge-Kutta method is unnecessary as well. A third-order k-exact finite volume GKS is also constructed with the help of k-exact reconstruction. Compared to the k-exact GKS, SCFV-GKS is compact, and with less computational cost for the reconstruction which is based on the main cell. Besides, a continuous reconstruction can be adopted in SCFV-GKS among subcells in smooth flow regions, which results in less numerical dissipation and less computational cost for flux evolution. Thus the efficiency of SCFV-GKS is much higher. Numerical tests demonstrate the strong robustness, high accuracy and efficiency of SCFV-GKS in a wide range of flow problems from nearly incompressible to supersonic flows with strong shock waves.

Modified Path-conservative HLLEM Scheme for Magnetohydrodynamic Solar Wind Simulations

Abstract The goal of the present work is to solve the magnetohydrodynamic (MHD) system of extended generalized Lagrange multiplier (EGLM) formulation with Galilean invariance (G-EGLM MHD equations) through a modified path-conservative HLLEM finite-volume method. A second-order least-squares reconstruction with Venkatakrishnan limiter is employed for state variables, and a solenoidality-preserving condition is considered for the magnetic field with the purpose of magnetic divergence cleaning. The two-stage Runge–Kutta time-integration method is utilized to advance the MHD governing equations. Compared with the original path-conservative HLLEM method, the modified method in this paper is shock stable and is able to adjust the diffusion according to the smoothness of the physical flow so as to automatically apply more diffusion near strong shocks and less in smooth regions near rarefaction waves and at contact discontinuities. Meanwhile, it can be robustly defined in the low plasma-β region. After several tests of smooth Alfvén wave, strong Lax, odd–even perturbation, and blast-wave problems, the large-scale structures of the solar corona for Carrington Rotation 2185 are numerically modeled in a six-component grid system of spherical coordinates with input from a Carrington rotation synoptic map provided by the Helioseismic and Magnetic Imager. Numerical results show the model’s capability of producing a structured solar wind in agreement with the observations.

Adaptive beam clean-up of high power slab lasers using least-squares wavefront reconstruction algorithm with performance-based filtering

The edge effects of high-power slab lasers cause severe wavefront distortions with large amplitude and gradients, which is a great challenge for the adaptive optics systems for improving the beam qualities. The standard least-squares wavefront reconstruction method would inherently assign higher priorities to correcting the edge aberrations which contribute little to the laser power in the near diffraction-limit areas in the far-field, while the other aberrations are not well corrected. To solve the problems, we propose a least-squares reconstruction algorithm with performance-based filtering, in which the correction capability of the deformable mirror is analyzed in the spatial frequency domain, and then the input phase distortions are filtered accordingly. The uncorrectable components of input distortions are ignored so that the residuals of the correctable components could be further minimized. Experiments with a 20 kW Nd:YAG slab laser indicate that the proposed method is effective to improve the beam quality of high-power slab laser with edge effect.