We propose a new characteristic-based finite volume scheme combined with the method of Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction and characteristics, to solve shallow water equations. We ap...We propose a new characteristic-based finite volume scheme combined with the method of Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction and characteristics, to solve shallow water equations. We apply the scheme to simulate dam-break problems. A number of challenging test cases are considered, such as large depth differences even wet/dry bed. The numerical solutions well agree with the analytical solutions. The results demonstrate the desired accuracy, high-resolution and robustness of the presented scheme.展开更多
Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenv...Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.展开更多
On March 15 a year ago,the Chinese top leader proposed the Global Civilisation Initiative,emphasising the importance of common development and common prosperity for the world.Cultural exchanges,essentially,involve sha...On March 15 a year ago,the Chinese top leader proposed the Global Civilisation Initiative,emphasising the importance of common development and common prosperity for the world.Cultural exchanges,essentially,involve sharing by people from different countries and backgrounds,ideas,values,traditions and customs,and other knowledge about each other's countries.In our increasingly intertwined world,cultural exchanges help broaden our vision and enrich our perspective by exposing us to different traditions,values,beliefs.展开更多
Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of ...Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.展开更多
We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrah...We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes.We use the Lax-Friedrichs monotone flux as building blocks,third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh,and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials.Numerical examples are given to demonstrate stability and accuracy of the scheme.展开更多
An extension of slant Hankel operator,namely,the kth-orderλ-slant Hankel operator on the Lebesgue space L^(2)(T^(n)),where T is the unit circle and n≥1,a natural number,is described in terms of the solution of a sys...An extension of slant Hankel operator,namely,the kth-orderλ-slant Hankel operator on the Lebesgue space L^(2)(T^(n)),where T is the unit circle and n≥1,a natural number,is described in terms of the solution of a system of operator equations,which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator.The study is further lifted in Calkin algebra in terms of essentially kth-orderλ-slant Hankel operators on L^(2)(T^(n)).展开更多
Aimed at the computational aeroacoustics multi-scale problem of complex configurations discretized with multi-size mesh, the flux reconstruction method based on modified Weight Essentially Non-Oscillatory（WENO） sche...Aimed at the computational aeroacoustics multi-scale problem of complex configurations discretized with multi-size mesh, the flux reconstruction method based on modified Weight Essentially Non-Oscillatory（WENO） scheme is proposed at the interfaces of multi-block grids.With the idea of Dispersion-Relation-Preserving（DRP） scheme, different weight coefficients are obtained by optimization, so that it is in WENO schemes with various characteristics of dispersion and dissipation. On the basis, hybrid flux vector splitting method is utilized to intelligently judge the amplitude of the gap between grid interfaces. After the simulation and analysis of 1D convection equation with different initial conditions, modified WENO scheme is proved to be able to independently distinguish the gap amplitude and generate corresponding dissipation according to the grid resolution. Using the idea of flux reconstruction at grid interfaces, modified WENO scheme with increasing dissipation is applied at grid points, while DRP scheme with low dispersion and dissipation is applied at the inner part of grids. Moreover, Gauss impulse spread and periodic point sound source flow among three cylinders with multi-scale grids are carried out. The results show that the flux reconstruction method at grid interfaces is capable of dealing with Computational Aero Acoustics（CAA） multi-scale problems.展开更多
A Weighted Essentially Non-Oscillatory scheme(WENO) is a solution to hyperbolic conservation laws,suitable for solving high-density fluid interface instability with strong intermittency. These problems have a large an...A Weighted Essentially Non-Oscillatory scheme(WENO) is a solution to hyperbolic conservation laws,suitable for solving high-density fluid interface instability with strong intermittency. These problems have a large and complex flow structure. To fully utilize the computing power of High Performance Computing(HPC) systems, it is necessary to develop specific methodologies to optimize the performance of applications based on the particular system’s architecture. The Sunway TaihuLight supercomputer is currently ranked as the fastest supercomputer in the world. This article presents a heterogeneous parallel algorithm design and performance optimization of a high-order WENO on Sunway TaihuLight. We analyzed characteristics of kernel functions, and proposed an appropriate heterogeneous parallel model. We also figured out the best division strategy for computing tasks,and implemented the parallel algorithm on Sunway TaihuLight. By using access optimization, data dependency elimination, and vectorization optimization, our parallel algorithm can achieve up to 172× speedup on one single node, and additional 58× speedup on 64 nodes, with nearly linear scalability.展开更多
The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the sol...The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.展开更多
In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical ...In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical reaction,say zi,is between 0 and 1,withΣz_(i)=1.Due to the lack of maximum-principle,most of the previous bound-preserving technique cannot be applied directly.To preserve those bounds,we will use the positivity-preserving technique to all the zi'is and enforceΣz_(i)=1 by constructing conservative schemes,thanks to conservative time integrations and consistent numerical fluxes in the system.Moreover,detonation is an extreme singular mode of flame propagation in premixed gas,and the model contains a significant stiff source.It is well known that for hyperbolic equations with stiff source,the transition points in the numerical approximations near the shocks may trigger spurious shock speed,leading to wrong shock position.Intuitively,the high-order weighted essentially non-oscillatory(WENO)scheme,which can suppress oscillations near the discontinuities,would be a good choice for spatial discretization.However,with the nonlinear weights,the numerical fluxes are no longer“consistent”,leading to nonconservative numerical schemes and the bound-preserving technique does not work.Numerical experiments demonstrate that,without further numerical techniques such as subcell resolutions,the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.展开更多
A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization an...A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving （SSP） Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.展开更多
In this paper, three versions of WENO schemes WENO-JS, WENO-M and WENO-Z are used for one-dimensional detonation wave simulations with fifth order characteristic based spatial flux reconstruction. Numerical schemes fo...In this paper, three versions of WENO schemes WENO-JS, WENO-M and WENO-Z are used for one-dimensional detonation wave simulations with fifth order characteristic based spatial flux reconstruction. Numerical schemes for solving the system of hyperbolic conversation laws using the ZND analytical solution as initial condition are presented. Numerical simulations of one-dimensional detonation wave for both stable and unstable cases are performed. In the stable case with overdrive factor f = 1.8, the temporal histories of peak pressure of the detonation front computed by WENO-JS and WENO-Z reach the theoretical steady state. In comparison, the temporal history of peak pressure computed by the WENO-M scheme fails to reach and oscillates around the theoretical steady state. In the unstable cases with overdrive factors f = 1.6 and f = 1.3, the results of all WENO schemes agree well with each other as the resolution, defined as the number of grid points per half-length of reaction zone, increases. Furthermore, for overdrive factor f = 1.6, the grid convergence study demonstrates that the high order WENO schemes converge faster than other existing lower order schemes such as unsplit scheme, Roe＇s solver with minmod limiter and Roe＇s solver with superbee limiter in reaching the predicted peak pressure. For overdrive factor f = 1.3, the temporal history of peak pressure shows an increasingly chaotic behavior even at high resolution. In the case of overdrive factor f = 1.1, in accordance with theoretical studies, an explosion occurs and different WENO schemes leading to this explosion appear at slightly different times.展开更多
The numerical method which is based on flux difference splitting, LUdecomposition, and implicit high-resolution third-order Essentially Non-Oscillatory (ENO) scheme wasconstructed for the efficient computation of stea...The numerical method which is based on flux difference splitting, LUdecomposition, and implicit high-resolution third-order Essentially Non-Oscillatory (ENO) scheme wasconstructed for the efficient computation of steady state solution to three-dimensionalincompressible Navier-Stokes e-quations in general coordinates. The flowfields over underwateraxisymmetric bodies, full-appended axisymmetric bodies and axisymetric bodies with a ring-wing ductwere simulated. The method is proved to be capable of predicting the circumferential-mean velocitydistribution at model scale to the accuracy of around 3% of measured values, and of predicting somedetails of flow features, for example, the wake harmonics.展开更多
High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of th...High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.展开更多
Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Im...Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Implicit integration is quite straightforward for first-order schemes.High order schemes instead also need to control spurious oscillations,which requires limiting in space and time also in the linear case.We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor,which is used both to set up the nonlinear weights of a standard high order space reconstruction,and to achieve limiting in time.In this preliminary work,we concentrate on the case of a third-order scheme,based on diagonally implicit Runge Kutta(DIRK)integration in time and central weighted essentially non-oscillatory(CWENO)reconstruction in space.The numerical tests involve linear and nonlinear scalar conservation laws.展开更多
In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed...In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.展开更多
To solve conservation laws,efficient schemes such as essentially nonoscillatory(ENO)and weighted ENO(WENO)have been introduced to control the Gibbs oscillations.Based on radial basis functions(RBFs)with the classical ...To solve conservation laws,efficient schemes such as essentially nonoscillatory(ENO)and weighted ENO(WENO)have been introduced to control the Gibbs oscillations.Based on radial basis functions(RBFs)with the classical WENO-JS weights,a new type of WENO schemes has been proposed to solve conservation laws[J.Guo et al.,J.Sci.Comput.,70(2017),pp.551–575].The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws.Unlike the usual method for reconstructing the flux functions,the flux function is generated directly with the conservative variables.Comparing with Guo and Jung(2017),the main advantage of this framework is that arbitrary monotone fluxes can be employed,while in Guo and Jung(2017)only smooth flux splitting can be used to reconstruct flux functions.Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.展开更多
Ellipsometry is a powerful method for determining both the optical constants and thickness of thin films.For decades,solutions to ill-posed inverse ellipsometric problems require substantial human-expert intervention ...Ellipsometry is a powerful method for determining both the optical constants and thickness of thin films.For decades,solutions to ill-posed inverse ellipsometric problems require substantial human-expert intervention and have become essentially human-in-the-loop trial-and-error processes that are not only tedious and time-consuming but also limit the applicability of ellipsometry.Here,we demonstrate a machine learning based approach for solving ellipsometric problems in an unambiguous and fully automatic manner while showing superior performance.The proposed approach is experimentally validated by using a broad range of films covering categories of metals,semiconductors,and dielectrics.This method is compatible with existing ellipsometers and paves the way for realizing the automatic,rapid,high-throughput optical characterization of films.展开更多
基金supported by the National Natural Science Foundation of China (Grant No.10771134)the Natural Science Foundation of Anhui Province (Grant No. 090416227)
文摘We propose a new characteristic-based finite volume scheme combined with the method of Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction and characteristics, to solve shallow water equations. We apply the scheme to simulate dam-break problems. A number of challenging test cases are considered, such as large depth differences even wet/dry bed. The numerical solutions well agree with the analytical solutions. The results demonstrate the desired accuracy, high-resolution and robustness of the presented scheme.
基金This work was done during the first authors＇ postdoctoral period in Qufu Normal University. This work was supported in part by the National Natural Science Foundation of China （Grant Nos. 11601261, 11671228） and the Natural Science Foundation of Shandong Province （No. ZR2016AQ12）.
文摘Finding the minimal H-eigenvalue of tensors is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a sum-of-squares （SOS） algorithm for computing the minimal H-eigenvalues of tensors with some sign structures called extended essentially nonnegative tensors （EEN-tensors）, which includes nonnegative tensors as a subclass. In the even-order symmetric case, we first discuss the positive semi-definiteness of EEN-tensors, and show that a positive semi-definite EEN-tensor is a non- negative tensor or an M-tensor or the sum of a nonnegative tensor and an M-tensor, then we establish a checkable sufficient condition for the SOS decomposition of EEN-tensors. Finally, we present an efficient algorithm to compute the minimal H-eigenvalues of even-order symmetric EEN-tensors based on the SOS decomposition. Numerical experiments are given to show the efficiency of the proposed algorithm.
文摘On March 15 a year ago,the Chinese top leader proposed the Global Civilisation Initiative,emphasising the importance of common development and common prosperity for the world.Cultural exchanges,essentially,involve sharing by people from different countries and backgrounds,ideas,values,traditions and customs,and other knowledge about each other's countries.In our increasingly intertwined world,cultural exchanges help broaden our vision and enrich our perspective by exposing us to different traditions,values,beliefs.
文摘Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady-state solutions of hyperbolic partial differential equations(PDEs).As other types of fast sweeping schemes,fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order.The resulting iterative schemes have a fast convergence rate to steady-state solutions.Moreover,an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve the inverse operation of any nonlinear local system.Hence,they are robust and flexible,and have been combined with high-order accurate weighted essentially non-oscillatory(WENO)schemes to solve various hyperbolic PDEs in the literature.For multidimensional nonlinear problems,high-order fixed-point fast sweeping WENO methods still require quite a large amount of computational costs.In this technical note,we apply sparse-grid techniques,an effective approximation tool for multidimensional problems,to fixed-point fast sweeping WENO methods for reducing their computational costs.Here,we focus on fixed-point fast sweeping WENO schemes with third-order accuracy(Zhang et al.2006[41]),for solving Eikonal equations,an important class of static Hamilton-Jacobi(H-J)equations.Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse-grid computations of the fixed-point fast sweeping WENO schemes achieve large savings of CPU times on refined meshes,and at the same time maintain comparable accuracy and resolution with those on corresponding regular single grids.
基金The research of the second author is supported by NSF grants AST-0506734 and DMS-0510345.
文摘We extend the weighted essentially non-oscillatory(WENO)schemes on two dimensional triangular meshes developed in[7]to three dimensions,and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes.We use the Lax-Friedrichs monotone flux as building blocks,third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh,and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials.Numerical examples are given to demonstrate stability and accuracy of the scheme.
文摘An extension of slant Hankel operator,namely,the kth-orderλ-slant Hankel operator on the Lebesgue space L^(2)(T^(n)),where T is the unit circle and n≥1,a natural number,is described in terms of the solution of a system of operator equations,which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator.The study is further lifted in Calkin algebra in terms of essentially kth-orderλ-slant Hankel operators on L^(2)(T^(n)).
文摘Aimed at the computational aeroacoustics multi-scale problem of complex configurations discretized with multi-size mesh, the flux reconstruction method based on modified Weight Essentially Non-Oscillatory（WENO） scheme is proposed at the interfaces of multi-block grids.With the idea of Dispersion-Relation-Preserving（DRP） scheme, different weight coefficients are obtained by optimization, so that it is in WENO schemes with various characteristics of dispersion and dissipation. On the basis, hybrid flux vector splitting method is utilized to intelligently judge the amplitude of the gap between grid interfaces. After the simulation and analysis of 1D convection equation with different initial conditions, modified WENO scheme is proved to be able to independently distinguish the gap amplitude and generate corresponding dissipation according to the grid resolution. Using the idea of flux reconstruction at grid interfaces, modified WENO scheme with increasing dissipation is applied at grid points, while DRP scheme with low dispersion and dissipation is applied at the inner part of grids. Moreover, Gauss impulse spread and periodic point sound source flow among three cylinders with multi-scale grids are carried out. The results show that the flux reconstruction method at grid interfaces is capable of dealing with Computational Aero Acoustics（CAA） multi-scale problems.
基金supported by the National High-Tech Research and Development (863) Program of China (No. 2015AA015306)the Science and Technology Plan of Beijing Municipality (No. Z161100000216147)+2 种基金the National Natural Science Foundation of China (No. 61762074)Youth Foundation Program of Qinghai University (No. 2016-QGY-5)the National Natural Science Foundation of Qinghai Province (No. 2019-ZJ7034)
文摘A Weighted Essentially Non-Oscillatory scheme(WENO) is a solution to hyperbolic conservation laws,suitable for solving high-density fluid interface instability with strong intermittency. These problems have a large and complex flow structure. To fully utilize the computing power of High Performance Computing(HPC) systems, it is necessary to develop specific methodologies to optimize the performance of applications based on the particular system’s architecture. The Sunway TaihuLight supercomputer is currently ranked as the fastest supercomputer in the world. This article presents a heterogeneous parallel algorithm design and performance optimization of a high-order WENO on Sunway TaihuLight. We analyzed characteristics of kernel functions, and proposed an appropriate heterogeneous parallel model. We also figured out the best division strategy for computing tasks,and implemented the parallel algorithm on Sunway TaihuLight. By using access optimization, data dependency elimination, and vectorization optimization, our parallel algorithm can achieve up to 172× speedup on one single node, and additional 58× speedup on 64 nodes, with nearly linear scalability.
基金Project supported by the National Natural Science Foundation of China(Nos.12072246,11972272,11872286)the National Numerical Wind Tunnel Project of China(No.NNW2020ZT3-A23)。
文摘The weakly ionized plasma flows in aerospace are commonly simulated by the single-fluid model,which cannot describe certain nonequilibrium phenomena by finite collisions of particles,decreasing the fidelity of the solution.Based on an alternative formulation of the targeted essentially non-oscillatory(TENO)scheme,a novel high-order numerical scheme is proposed to simulate the two-fluid plasmas problems.The numerical flux is constructed by the TENO interpolation of the solution and its derivatives,instead of being reconstructed from the physical flux.The present scheme is used to solve the two sets of Euler equations coupled with Maxwell's equations.The numerical methods are verified by several classical plasma problems.The results show that compared with the original TENO scheme,the present scheme can suppress the non-physical oscillations and reduce the numerical dissipation.
基金the National Natural Science Foundation of China under Grant Number NSFC 11801302Tsinghua University Initiative Scientific Research Program.Yang Yang is supported by the NSF Grant DMS-1818467.
文摘In this paper,we apply high-order finite difference(FD)schemes for multispecies and multireaction detonations(MMD).In MMD,the density and pressure are positive and the mass fraction of the ith species in the chemical reaction,say zi,is between 0 and 1,withΣz_(i)=1.Due to the lack of maximum-principle,most of the previous bound-preserving technique cannot be applied directly.To preserve those bounds,we will use the positivity-preserving technique to all the zi'is and enforceΣz_(i)=1 by constructing conservative schemes,thanks to conservative time integrations and consistent numerical fluxes in the system.Moreover,detonation is an extreme singular mode of flame propagation in premixed gas,and the model contains a significant stiff source.It is well known that for hyperbolic equations with stiff source,the transition points in the numerical approximations near the shocks may trigger spurious shock speed,leading to wrong shock position.Intuitively,the high-order weighted essentially non-oscillatory(WENO)scheme,which can suppress oscillations near the discontinuities,would be a good choice for spatial discretization.However,with the nonlinear weights,the numerical fluxes are no longer“consistent”,leading to nonconservative numerical schemes and the bound-preserving technique does not work.Numerical experiments demonstrate that,without further numerical techniques such as subcell resolutions,the conservative FD method with linear weights can yield better numerical approximations than the nonconservative WENO scheme.
基金Project supported by the National Natural Science Foundation of China (Grant No: 60134010).
文摘A numerical two-dimensional shallow water method was based on method for solving the equations was presented. This the third-order genuinely multidimensional semi-discrete central scheme for spatial discretization and the optimal third-order Strong Stability Preserving （SSP） Runge-Kutta method for time integration. The third-order compact Central Weighted Essentially Non-Oscillatory （CWENO） reconstruction was adopted to guarantee the non-oscillatory behavior of the presented scheme and improve the resolution. Two kinds of source terms were considered in this work. They were evaluated using different approaches. The resulting scheme does not require Riemann solvers or characteristic decomposition, hence it retains all the attractive features of central schemes such as simplicity and high resolution. To evaluate the performance of the presented scheme, several numerical examples were tested. The results demonstrate that our method is efficient, stable and robust.
文摘In this paper, three versions of WENO schemes WENO-JS, WENO-M and WENO-Z are used for one-dimensional detonation wave simulations with fifth order characteristic based spatial flux reconstruction. Numerical schemes for solving the system of hyperbolic conversation laws using the ZND analytical solution as initial condition are presented. Numerical simulations of one-dimensional detonation wave for both stable and unstable cases are performed. In the stable case with overdrive factor f = 1.8, the temporal histories of peak pressure of the detonation front computed by WENO-JS and WENO-Z reach the theoretical steady state. In comparison, the temporal history of peak pressure computed by the WENO-M scheme fails to reach and oscillates around the theoretical steady state. In the unstable cases with overdrive factors f = 1.6 and f = 1.3, the results of all WENO schemes agree well with each other as the resolution, defined as the number of grid points per half-length of reaction zone, increases. Furthermore, for overdrive factor f = 1.6, the grid convergence study demonstrates that the high order WENO schemes converge faster than other existing lower order schemes such as unsplit scheme, Roe＇s solver with minmod limiter and Roe＇s solver with superbee limiter in reaching the predicted peak pressure. For overdrive factor f = 1.3, the temporal history of peak pressure shows an increasingly chaotic behavior even at high resolution. In the case of overdrive factor f = 1.1, in accordance with theoretical studies, an explosion occurs and different WENO schemes leading to this explosion appear at slightly different times.
文摘The numerical method which is based on flux difference splitting, LUdecomposition, and implicit high-resolution third-order Essentially Non-Oscillatory (ENO) scheme wasconstructed for the efficient computation of steady state solution to three-dimensionalincompressible Navier-Stokes e-quations in general coordinates. The flowfields over underwateraxisymmetric bodies, full-appended axisymmetric bodies and axisymetric bodies with a ring-wing ductwere simulated. The method is proved to be capable of predicting the circumferential-mean velocitydistribution at model scale to the accuracy of around 3% of measured values, and of predicting somedetails of flow features, for example, the wake harmonics.
文摘High-order accurate weighted essentially non-oscillatory(WENO)schemes are a class of broadly applied numerical methods for solving hyperbolic partial differential equations(PDEs).Due to highly nonlinear property of the WENO algorithm,large amount of computational costs are required for solving multidimensional problems.In our previous work(Lu et al.in Pure Appl Math Q 14:57–86,2018;Zhu and Zhang in J Sci Comput 87:44,2021),sparse-grid techniques were applied to the classical finite difference WENO schemes in solving multidimensional hyperbolic equations,and it was shown that significant CPU times were saved,while both accuracy and stability of the classical WENO schemes were maintained for computations on sparse grids.In this technical note,we apply the approach to recently developed finite difference multi-resolution WENO scheme specifically the fifth-order scheme,which has very interesting properties such as its simplicity in linear weights’construction over a classical WENO scheme.Numerical experiments on solving high dimensional hyperbolic equations including Vlasov based kinetic problems are performed to demonstrate that the sparse-grid computations achieve large savings of CPU times,and at the same time preserve comparable accuracy and resolution with those on corresponding regular single grids.
基金MIUR(Ministry of University and Research)PRIN2017 project number 2017KKJP4XProgetto di Ateneo Sapienza,number RM120172B41DBF3A.
文摘Many interesting applications of hyperbolic systems of equations are stiff,and require the time step to satisfy restrictive stability conditions.One way to avoid small time steps is to use implicit time integration.Implicit integration is quite straightforward for first-order schemes.High order schemes instead also need to control spurious oscillations,which requires limiting in space and time also in the linear case.We propose a framework to simplify considerably the application of high order non-oscillatory schemes through the introduction of a low order implicit predictor,which is used both to set up the nonlinear weights of a standard high order space reconstruction,and to achieve limiting in time.In this preliminary work,we concentrate on the case of a third-order scheme,based on diagonally implicit Runge Kutta(DIRK)integration in time and central weighted essentially non-oscillatory(CWENO)reconstruction in space.The numerical tests involve linear and nonlinear scalar conservation laws.
基金supported by the NSF (Grant No.DMS-1753581)supported by NSFC (Grant No.12071392).
文摘In this paper,we propose a novel Hermite weighted essentially non-oscillatory(HWENO)fast sweeping method to solve the static Hamilton-Jacobi equations efficiently.During the HWENO reconstruction procedure,the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils.However,one major novelty and difference from the traditional HWENO framework lies in the fact that,we do not need to introduce and solve any additional equations to update the derivatives of the unknown functionϕ.Instead,we use the currentϕand the old spatial derivative ofϕto update them.The traditional HWENO fast sweeping method is also introduced in this paper for comparison,where additional equations governing the spatial derivatives ofϕare introduced.The novel HWENO fast sweeping methods are shown to yield great savings in computational time,which improves the computational efficiency of the traditional HWENO scheme.In addition,a hybrid strategy is also introduced to further reduce computational costs.Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
文摘To solve conservation laws,efficient schemes such as essentially nonoscillatory(ENO)and weighted ENO(WENO)have been introduced to control the Gibbs oscillations.Based on radial basis functions(RBFs)with the classical WENO-JS weights,a new type of WENO schemes has been proposed to solve conservation laws[J.Guo et al.,J.Sci.Comput.,70(2017),pp.551–575].The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws.Unlike the usual method for reconstructing the flux functions,the flux function is generated directly with the conservative variables.Comparing with Guo and Jung(2017),the main advantage of this framework is that arbitrary monotone fluxes can be employed,while in Guo and Jung(2017)only smooth flux splitting can be used to reconstruct flux functions.Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.
基金the National Key R&D Program of China(2017YFA0305100,2017YFA0303800,and 2019YFA0705000)National Natural Science Foundation of China(92050114,62076140,91750204,61775106,11904182,61633012,11711530205,11374006,12074200,and 11774185)+6 种基金Guangdong Major Project of Basic and Applied Basic Research(2020B0301030009)111 Projea(B07013)PCSIRT(IRT0149)Open Research Program of Key Laboratory of 3D Micro/Nano Fabrication and Characterization of Zhejiang ProvinceTianjin Youth Talent Support ProgramFundamental Research Funds for the Central Universities(010-63201003,010-63201008,and 010-63201009)。
文摘Ellipsometry is a powerful method for determining both the optical constants and thickness of thin films.For decades,solutions to ill-posed inverse ellipsometric problems require substantial human-expert intervention and have become essentially human-in-the-loop trial-and-error processes that are not only tedious and time-consuming but also limit the applicability of ellipsometry.Here,we demonstrate a machine learning based approach for solving ellipsometric problems in an unambiguous and fully automatic manner while showing superior performance.The proposed approach is experimentally validated by using a broad range of films covering categories of metals,semiconductors,and dielectrics.This method is compatible with existing ellipsometers and paves the way for realizing the automatic,rapid,high-throughput optical characterization of films.
