A Class of Hybrid DG/FV Methods for Conservation Laws III: Two-Dimensional Euler Equations

A concept of "static reconstruction" and "dynamic reconstruction" was introduced for higher-order (third-order or more) numerical methods in our previous work. Based on this concept, a class of hybrid DG/FV methods had been developed for one-dimensional conservation law using a "hybrid reconstruction" approach, and extended to two-dimensional scalar equations on triangular and Cartesian/triangular hybrid grids. In the hybrid DG/FV schemes, the lower-order derivatives of the piecewise polynomial are computed locally in a cell by the traditional DG method (called as "dynamic reconstruction"), while the higher-order derivatives are reconstructed by the "static reconstruction" of the FV method, using the known lower-order derivatives in the cell itself and in its adjacent neighboring cells. In this paper, the hybrid DG/FV schemes are extended to two-dimensional Euler equations on triangular and Cartesian/triangular hybrid grids. Some typical test cases are presented to demonstrate the performance of the hybrid DG/FV methods, including the standard vortex evolution problem with exact solution, isentropic vortex/weak shock wave interaction, subsonic flows past a circular cylinder and a three-element airfoil (30P30N), transonic flow past a NACA0012 airfoil. The accuracy study shows that the hybrid DG/FV method achieves the desired third-order accuracy, and the applications demonstrate that they can capture the flow structure accurately, and can reduce the CPU time and memory requirement greatly than the traditional DG method with the same order of accuracy.     

An Implicit Algorithm for High-Order DG/FV Schemes for Compressible Flows on 2D Arbitrary Grids

A Newton/LU-SGS (lower-upper symmetric Gauss-Seidel) iteration implicit method was developed to solve two-dimensional Euler and Navier-Stokes equations by the DG/FV hybrid schemes on arbitrary grids. The Newton iteration was employed to solve the nonlinear system, while the linear system was solved with LU-SGS iteration. The effect of several parameters in the implicit scheme, such as the CFL number, the Newton sub-iteration steps, and the update frequency of Jacobian matrix, was investigated to evaluate the performance of convergence history. Several typical test cases were simulated, and compared with the traditional explicit Runge-Kutta (RK) scheme. Firstly the Couette flow was tested to validate the order of accuracy of the present DG/FV hybrid schemes. Then a subsonic inviscid flow over a bump in a channel was simulated and the effect of parameters was alsoinvestigated. Finally, the implicit algorithm was applied to simulate a subsonic inviscid flow over a circular cylinder and the viscous flow in a square cavity. The numerical results demonstrated that the present implicit scheme can accelerate the convergence history efficiently. Choosing proper parameters would improve the efficiency of the implicit scheme. Moreover, in the same framework, the DG/FV hybrid schemes are more efficient than the same order DG schemes.     

The Error-Minimization-Based Strategy for Moving Mesh Methods

The typical elements in a numerical simulation of fluid flow using moving meshes are a time integration scheme, a rezone method in which a new mesh is defined, and a remapping (conservative interpolation) in which a solution is transferred to the new mesh. The objective of the rezone method is to move the computational mesh to improve the robustness, accuracy and eventually efficiency of the simulation. In this paper, we consider the one-dimensional viscous Burgers' equation and describe a new rezone strategy which minimizes the L₂ norm of error and maintains mesh smoothness. The efficiency of the proposed method is demonstrated with numerical examples.     

A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

In this paper, we investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively. The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities. In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique not only reduces the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency. The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given grid resolution. A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements.     

Finite-Volume-Particle Methods for the Two-Component Camassa-Holm System

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.     

A Higher Order Interpolation Scheme of Finite Volume Method for Compressible Flow on Curvilinear Grids

A higher order interpolation scheme based on a multi-stage BVD (Boundary Variation Diminishing) algorithm is developed for the FV (Finite Volume) method on non-uniform, curvilinear structured grids to simulate the compressible turbulent flows. The designed scheme utilizes two types of candidate interpolants including a higher order linear-weight polynomial as high as eleven and a THINC (Tangent of Hyperbola for INterface Capturing) function with the adaptive steepness. We investigate not only the accuracy but also the efficiency of the methodology through the cost efficiency analysis in comparison with well-designed mapped WENO (Weighted Essentially Non-Oscillatory) scheme. Numerical experimentation including benchmark broadband turbulence problem as well as real-life wall-bounded turbulent flows has been carried out to demonstrate the potential implementation of the present higher order interpolation scheme especially in the ILES (Implicit Large Eddy Simulation) of compressible turbulence.     

腰椎退行性滑脱与棘间韧带MRI的T2WI高信号相关性分析

目的:探讨腰椎退行性滑脱与MRI棘间韧带T2WI高信号之间的关系,以提高对棘间韧带信号改变的认识。方法:收集2018年3月至2020年3月临床诊断为腰椎退行性滑脱43例患者的MRI资料,男19例,女24例,年龄50~92岁,平均69岁。利用影像归档和通信系统(picture archiving and communication systems,PACS)调阅影像,记录滑脱节段与非滑脱节段棘间韧带出现T2WI高信号的分布情况和发生率,利用Spearman分析棘间韧带的T2WI高信号与腰椎滑脱程度的关系。结果:除8条韧带因图像显示不佳未计入统计结果外,43例患者共207个腰椎椎体和相应的棘间韧带入组研究。根据Meyerding分型法,43例患者共有48个节段出现滑脱,Ⅰ度滑脱41个节段,Ⅱ度滑脱7个节段。滑脱节段对应的棘间韧带出现T2WI高信号30例,其中L_2,3节段3例,L_3,4节段3例,L_4,5节段20例,L5S1节段4例;159个非滑脱节段对应的棘间韧带出现T2WI高信号53例,其中L1,2节段6例,L_2,3节段6例,L_3,4节段13例,L_4,5节段7例,L₅S₁节段21例。滑脱节段与非滑脱节段相比,棘间韧带T2WI高信号的发生率分别为62.5%和33.3%,差异有统计学意义(χ²=13.06,P<0.05)。Spearman相关分析显示棘间韧带T2WI高信号的出现与腰椎滑脱程度呈正相关(r=0.264,P<0.05)。结论:退行性腰椎滑脱患者中滑脱椎体的棘间韧带出现T2WI高信号更多见,T2WI高信号的出现与椎体滑脱的程度呈正相关,在影像诊断中应引起足够重视。     

Modic改变与下腰椎三关节复合体退变相关性的回顾性分析

目的探讨Modic改变(modic changes,MCs)与下腰椎三关节复合体退变的相关性。方法选择2016年3月~2020年6月在本院住院治疗的231例腰椎间盘突出症(lumbar disc herniation,LDH)患者进行分析,观察MCs的发生率、椎间盘的Pfirmann分级和小关节退变分级(Weishaupt分级)的关系。结果MCs总发生率为45.31%(296/693),L_3-4、L_4-5、L₅-S₁节段MCs发生率分别为25.11%(58/231)、54.11%(125/231)和48.92%(113/231),组间差异存在统计学意义(P<0.05)。L_3-4、L_4-5和L₅-S₁节段MCsⅠ型、Ⅱ型和Ⅲ型病变节段的椎间盘退变程度均高于无MCs病变节段(P<0.05)。L_3-4节段MCsⅢ型与无MCs患者的小关节退变差异存在统计学意义(P<0.05)。L_4-5和L₅-S₁节段MCsⅡ型患者与无MCs患者的小关节退变差异存在统计学意义(P<0.05)。结论MCs与三关节复合体退变存在相关性,主要表现在MCs不同类型均与腰椎间盘退变分级相关,MCsⅡ型与腰椎小关节退行性病变相关。     

A Slope Constrained 4th Order Multi-Moment Finite Volume Method with WENO Limiter

This paper presents a new and better suited formulation to implement the limiting projection to high-order schemes that make use of high-order local reconstructions for hyperbolic conservation laws. The scheme, so-called MCV-WENO4 (multi-moment Constrained finite Volume with WENO limiter of 4th order) method, is an extension of the MCV method of Ii & Xiao (2009) by adding the 1st order derivative (gradient or slope) at the cell center as an additional constraint for the cell-wise local reconstruction. The gradient is computed from a limiting projection using the WENO (weighted essentially non-oscillatory) reconstruction that is built from the nodal values at 5 solution points within 3 neighboring cells. Different from other existing methods where only the cell-average value is used in the WENO reconstruction, the present method takes account of the solution structure within each mesh cell, and thus minimizes the stencil for reconstruction. The resulting scheme has 4th-order accuracy and is of significant advantage in algorithmic simplicity and computational efficiency. Numerical results of one and two dimensional benchmark tests for scalar and Euler conservation laws are shown to verify the accuracy and oscillation-less property of the scheme.     

An Implicit LU-SGS Scheme for the Spectral Volume Method on Unstructured Tetrahedral Grids

An efficient implicit lower-upper symmetric Gauss-Seidel (LU-SGS) solution approach has been applied to a high order spectral volume (SV) method for unstructured tetrahedral grids. The LU-SGS solver is preconditioned by the block element matrix, and the system of equations is then solved with a LU decomposition. The compact feature of SV reconstruction facilitates the efficient solution algorithm even for high order discretizations. The developed implicit solver has shown more than an order of magnitude of speed-up relative to the Runge-Kutta explicit scheme for typical inviscid and viscous problems. A convergence to a high order solution for high Reynolds number transonic flow over a 3D wing with a one equation turbulence model is also indicated.     

The predictive value of quantitative computed tomography for vertebral body compressive strength and ash density

Whole lumbar vertebral sections (L₂ and L₃) were obtained from 30 elderly individuals aged 43–95 years, mean 81 years (13 females, 17 males). None of the subjects had had malignant diseases. Quantitative computed tomography (QCT) was performed on an EMI 7070 scanner. One 8 mm slice parallel to the end-plates was obtained from the center of each vertebral body. The trabecular bone mass in each slice was outlined interactively by means of a tracer-ball. A CT-histogram was recorded inside this area, and average CT-values were expressed in Hounsfield Units (HU).

The whole vertebral body (L₂) was compressed in a materials testing machine. From the central part of L₃, vertical cylindrical pure trabecular bone specimens were obtained. The biomechanical competence of these specimens was also assessed by means of a materials testing machine. Finally, all bone specimens were incinerated for determination of apparent ash-density.

Highly significant positive correlations were found between average CT-values and (a) stress values of the trabecular bone (r = 0.81, p < 0.001) and (b) ash-density of the pure trabecular bone (r = 0.81, p < 0.001). Furthermore, a significant positive correlation was found between CT-values and (a) total vertebral body load (r = 0.72, p < 0.001), (b) total vertebral body stress (load/cross-sectional area) (r = 0.55, p < 0.001) and (c) ash-density of the whole vertebral body (r = 0.76, p < 0.001).

It is concluded that quantitative computed tomography gives valid predictions of both vertebral trabecular bone mass and mechanical competence. The predictive value for whole vertebral body load, stress and ash-density, although less marked, is still highly significant.     

High-Order Runge-Kutta Discontinuous Galerkin Methods with a New Type of Multi-Resolution WENO Limiters on Tetrahedral Meshes

In this paper, the second-order and third-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are proposed on tetrahedral meshes. The multi-resolution WENO limiter is an extension of a finite volume multi-resolution WENO scheme developed in [81], which serves as a limiter for RKDG methods on tetrahedral meshes. This new WENO limiter uses information of the DG solution essentially only within the troubled cell itself which is identified by a new modified version of the original KXRCF indicator [42], to build a sequence of hierarchical $L^2$ projection polynomials from zeroth degree to the second or third degree of the DG solution. The second-order and third-order RKDG methods with the associated multi-resolution WENO limiters are developed as examples for general high-order RKDG methods, which could maintain the original order of accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities by gradually degrading from the optimal order to the first order. The linear weights inside the procedure of the new multi-resolution WENO limiters can be set as any positive numbers on the condition that they sum to one. A series of polynomials of different degrees within the troubled cell itself are applied in a WENO fashion to modify the DG solutions in the troubled cell on tetrahedral meshes. These new WENO limiters are very simple to construct, and can be easily implemented to arbitrary high-order accuracy on tetrahedral meshes. Such spatial reconstruction methodology improves the robustness in the simulation on the same compact spatial stencil of the original DG methods on tetrahedral meshes. Extensive one-dimensional (run as three-dimensional problems on tetrahedral meshes) and three-dimensional tests are performed to demonstrate the good performance of the RKDG methods with new multi-resolution WENO limiters.     

A DDG Method with a Residual-Based Artificial Viscosity for the Transonic/Supersonic Compressible Flow

In this work, a direct discontinuous Galerkin (DDG) method with artificial viscosity is developed to solve the compressible Navier-Stokes equations for simulating the transonic or supersonic flow, where the DDG approach is used to discretize viscous and heat fluxes. A strong residual-based artificial viscosity (AV) technique is proposed to be applied in the DDG framework to handle shock waves and layer structures appearing in transonic or supersonic flow, which promotes convergence and robustness. Moreover, the AV term is added to classical BR2 methods for comparison. A number of 2-D and 3-D benchmarks such as airfoils, wings, and a full aircraft are presented to assess the performance of the DDG framework with the strong residual-based AV term for solving the two dimensional and three dimensional Navier-Stokes equations. The proposed framework provides an alternative robust and efficient approach for numerically simulating the multi-dimensional compressible Navier-Stokes equations for transonic or supersonic flow.     

MR引导经中后与侧后方入路腰椎间盘穿刺方法对照

目的与经侧后方入路比较,探讨MR引导下经中后方入路腰椎间盘穿刺的可行性及优势。方法在配有iPath200光学导航系统的0.23T开放式介入磁共振系统实时引导下,对需要介入治疗的73例腰椎间盘突出患者、共计116个椎间盘分别经侧后方和中后方入路进行穿刺。由2名MR诊断医师使用术中预扫描图像对拟穿刺椎间盘的退变程度共同进行评价。根据穿刺成功所需穿刺次数、操作时间及并发症出现的例数,对两种穿刺方法进行综合评价。结果 31例患者的49个腰椎间盘经侧后方入路穿刺,42例患者的67个腰椎间盘经中后方入路穿刺,所有病例经两种方法均成功穿刺。两组患者椎间盘退变程度差异无统计学意义。两种入路成功穿刺L3-4、L4-5椎间盘所需穿刺次数、操作时间差异无统计学意义（P均〉0.05）;经中后方入路成功穿刺L5-S1椎间盘所需穿刺次数[（1.43±0.68）次vs（2.14±0.77）次]和时间[（9.02±2.50）min vs（14.61±3.93）min]均显著低于经侧后方入路（P均〈0.05）;两种穿刺入路短期并发症差异无统计学意义（P〉0.05）。结论 MR引导下经中后方入路穿刺腰椎间盘较与侧后方入路同样安全、可行。对于L5-S1椎间盘,经中后方入路穿刺较侧后方入路可明显减少穿刺次数,缩短操作时间。     

Conservative Semi-Lagrangian Finite Difference WENO Formulations with Applications to the Vlasov Equation

In this paper, we propose a new conservative semi-Lagrangian (SL) finite difference (FD) WENO scheme for linear advection equations, which can serve as a base scheme for the Vlasov equation by Strang splitting [4]. The reconstruction procedure in the proposed SL FD scheme is the same as the one used in the SL finite volume (FV) WENO scheme [3]. However, instead of inputting cell averages and approximate the integral form of the equation in a FV scheme, we input point values and approximate the differential form of equation in a FD spirit, yet retaining very high order (fifth order in our experiment) spatial accuracy. The advantage of using point values, rather than cell averages, is to avoid the second order spatial error, due to the shearing in velocity (v) and electrical field (E) over a cell when performing the Strang splitting to the Vlasov equation. As a result, the proposed scheme has very high spatial accuracy, compared with second order spatial accuracy for Strang split SL FV scheme for solving the Vlasov-Poisson (VP) system. We perform numerical experiments on linear advection, rigid body rotation problem; and on the Landau damping and two-stream instabilities by solving the VP system. For comparison, we also apply (1) the conservative SL FD WENO scheme, proposed in [22] for incompressible advection problem, (2) the conservative SL FD WENO scheme proposed in [21] and (3) the non-conservative version of the SL FD WENO scheme in [3] to the same test problems. The performances of different schemes are compared by the error table, solution resolution of sharp interface, and by tracking the conservation of physical norms, energies and entropies, which should be physically preserved.     

A Multi-Material CCALE-MOF Approach in Cylindrical Geometry

In this paper we present recent developments concerning a Cell-Centered Arbitrary Lagrangian Eulerian (CCALE) strategy using the Moment Of Fluid (MOF) interface reconstruction for the numerical simulation of multi-material compressible fluid flows on unstructured grids in cylindrical geometries. Especially, our attention is focused here on the following points. First, we propose a new formulation of the scheme used during the Lagrangian phase in the particular case of axisymmetric geometries. Then, the MOF method is considered for multi-interface reconstruction in cylindrical geometry. Subsequently, a method devoted to the rezoning of polar meshes is detailed. Finally, a generalization of the hybrid remapping to cylindrical geometries is presented. These explorations are validated by mean of several test cases using unstructured grid that clearly illustrate the robustness and accuracy of the new method.     

Performance Enhancement for High-Order Gas-Kinetic Scheme Based on WENO-Adaptive-Order Reconstruction

High-order gas-kinetic scheme (HGKS) has been well-developed in the past years. Abundant numerical tests including hypersonic flow, turbulence, and aeroacoustic problems, have been used to validate its accuracy, efficiency, and robustness. However, there is still room for its further improvement. Firstly, the reconstruction in the previous scheme mainly achieves a fifth-order accuracy for the point-wise values at a cell interface due to the use of standard WENO reconstruction, and the slopes of the initial non-equilibrium states have to be reconstructed from the cell interface values and cell averages again. The same order of accuracy for slopes as the original WENO scheme cannot be achieved. At the same time, the equilibrium state in space and time in HGKS has to be reconstructed separately. Secondly, it is complicated to get reconstructed data at Gaussian points from the WENO-type method in high dimensions. For HGKS, besides the point-wise values at the Gaussian points it also requires the slopes in both normal and tangential directions of a cell interface. Thirdly, there exists visible spurious overshoot/undershoot at weak discontinuities from the previous HGKS with the standard WENO reconstruction. In order to overcome these difficulties, in this paper we use an improved reconstruction for HGKS. The WENO with adaptive order (WENO-AO) [2] method is implemented for reconstruction. Equipped with WENO-AO reconstruction, the performance enhancement of HGKS is fully explored. WENO-AO not only provides the interface values, but also the slopes. In other words, a whole polynomial inside each cell is provided by the WENO-AO reconstruction. The available polynomial may not benefit to the high-order schemes based on the Riemann solver, where only points-wise values at the cell interface are needed. But, it can be fully utilized in the HGKS. As a result, the HGKS becomes simpler than the previous one with the direct implementation of cell interface values and their slopes from WENO-AO. The additional reconstruction of equilibrium state at the beginning of each time step can be avoided as well by dynamically merging the reconstructed non-equilibrium slopes. The new HGKS essentially releases or totally removes the above existing problems in the previous HGKS. The accuracy of the scheme from 1D to 3D from the new HGKS can recover the theoretical order of accuracy of the WENO reconstruction. In the two- and three-dimensional simulations, the new HGKS shows better robustness and efficiency than the previous scheme in all test cases.     

大通道脊柱内镜下单侧入路双侧减压治疗腰椎管狭窄症的临床疗效

目的 探讨大通道脊柱内镜下单侧入路双侧减压治疗腰椎管狭窄症(lumbar spinal stenosis,LSS)的临床疗效.方法 纳入2018年1月～2020年6月本院收治的75例LSS患者,其中男41例,女34例,年龄(66.59±4.97)岁,L3-412例,L4-540例,L5-S123例;均采用大通道脊柱内镜...     

椎板间入路经皮内镜椎间盘摘除术治疗L5~S1腋下型腰椎间盘突出症效果分析

目的探讨椎板间入路皮内镜经椎间盘摘除术(PEID)治疗L₅~S₁腋下型腰椎间盘突出症(LDH)的临床效果。方法回顾性分析2014-07—2020-06中国人民解放军联勤保障部队第九九〇医院信阳医疗区骨科收治的72例L₅~S₁腋下型LDH患者的临床资料。分为椎板间入路PEID组(PEID组,31例)和传统后路开放椎间盘摘除术组(传统开放组,41例)。比较2组患者的手术情况、术后临床指标及各时间点的视觉模拟评分(VAS)、Oswestry功能障碍指数(ODI)。结果PEID组的切口长度、手术时间、术中出血量,以及术后卧床时间和拆线时间均短(少)于传统开放组,差异有统计学意义(P<0.05)。术后随访9~12个月,2组患者各时间点的VAS、ODI评分均优于术前,差异有统计学意义(P<0.05)。除PEID组患者术后第3天的VAS评分优于传统开放组,差异有统计学意义(P<0.05)外,其他时间点2组患者的VAS、ODI评分,以及并发症发生率差异均无统计学意义(P>0.05)。结论椎板间入路PEID与传统后路开放椎间盘摘除术治疗L₅~S₁腋下型LDH,在改善患者的VAS、ODI评分方面差异均无统计学意义,但椎板间入路PEID具有切口小、手术时间短、术中出血量少、术后早期痛苦小,以及并发症发生率不高等优势,故更有利于患者术后恢复。     

Three-Dimensional Lattice Boltzmann Flux Solver and Its Applications to Incompressible Isothermal and Thermal Flows

A three-dimensional (3D) lattice Boltzmann flux solver (LBFS) is presented in this paper for the simulation of both isothermal and thermal flows. The present solver combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. It applies the finite volume method (FVM) to solve the N-S equations. Different from the conventional N-S solvers, its viscous and inviscid fluxes at the cell interface are evaluated simultaneously by local reconstruction of LBE solution. As compared to the conventional LBE solvers, which apply the lattice Boltzmann method (LBM) globally in the whole computational domain, it only applies LBM locally at each cell interface, and flow variables at cell centers are given from the solution of N-S equations. Since LBM is only applied locally in the 3D LBFS, the drawbacks of the conventional LBM, such as limitation to uniform mesh, tie-up of mesh spacing and time step, tedious implementation of boundary conditions, are completely removed. The accuracy, efficiency and stability of the proposed solver are examined in detail by simulating plane Poiseuille flow, lid-driven cavity flow and natural convection. Numerical results show that the LBFS has a second order of accuracy in space. The efficiency of the LBFS is lower than LBM on the same grids. However, the LBFS needs very less non-uniform grids to get grid-independence results and its efficiency can be greatly improved and even much higher than LBM. In addition, the LBFS is more stable and robust.