A high-order gas kinetic flux solver(GKFS)is presented for simulating inviscid compressible flows.The weighted essentially non-oscillatory(WENO)scheme on a uniform mesh in the finite volume formulation is combined wit...A high-order gas kinetic flux solver(GKFS)is presented for simulating inviscid compressible flows.The weighted essentially non-oscillatory(WENO)scheme on a uniform mesh in the finite volume formulation is combined with the circular function-based GKFS(C-GKFS)to capture more details of the flow fields with fewer grids.Different from most of the current GKFSs,which are constructed based on the Maxwellian distribution function or its equivalent form,the C-GKFS simplifies the Maxwellian distribution function into the circular function,which ensures that the Euler or Navier-Stokes equations can be recovered correctly.This improves the efficiency of the GKFS and reduces its complexity to facilitate the practical application of engineering.Several benchmark cases are simulated,and good agreement can be obtained in comparison with the references,which demonstrates that the high-order C-GKFS can achieve the desired accuracy.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
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
In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume fram...In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.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 novel third-order optimized symmetric weighted essentially non-oscillatory(WENO-OS3)scheme is used to simulate the hypersonic shock wave/boundary layer interactions.Firstly,the scheme is presented with the achieveme...A novel third-order optimized symmetric weighted essentially non-oscillatory(WENO-OS3)scheme is used to simulate the hypersonic shock wave/boundary layer interactions.Firstly,the scheme is presented with the achievement of low dissipation in smooth region and robust shock-capturing capabilities in discontinuities.The Maxwell slip boundary conditions are employed to consider the rarefied effect near the surface.Secondly,several validating tests are given to show the good resolution of the WENO-OS3 scheme and the feasibility of the Maxwell slip boundary conditions.Finally,hypersonic flows around the hollow cylinder truncated flare(HCTF)and the25°/55°sharp double cone are studied.Discussions are made on the characteristics of the hypersonic shock wave/boundary layer interactions with and without the consideration of the slip effect.The results indicate that the scheme has a good capability in predicting heat transfer with a high resolution for describing fluid structures.With the slip boundary conditions,the separation region at the corner is smaller and the prediction is more accurate than that with no-slip boundary conditions.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
基金Project supported by the National Natural Science Foundation of China(No.12072158)。
文摘A high-order gas kinetic flux solver(GKFS)is presented for simulating inviscid compressible flows.The weighted essentially non-oscillatory(WENO)scheme on a uniform mesh in the finite volume formulation is combined with the circular function-based GKFS(C-GKFS)to capture more details of the flow fields with fewer grids.Different from most of the current GKFSs,which are constructed based on the Maxwellian distribution function or its equivalent form,the C-GKFS simplifies the Maxwellian distribution function into the circular function,which ensures that the Euler or Navier-Stokes equations can be recovered correctly.This improves the efficiency of the GKFS and reduces its complexity to facilitate the practical application of engineering.Several benchmark cases are simulated,and good agreement can be obtained in comparison with the references,which demonstrates that the high-order C-GKFS can achieve the desired accuracy.
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.
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supported in part by the NSF grant DMS-2208438.The work of M.Herty was supported in part by the DFG(German Research Foundation)through 20021702/GRK2326,333849990/IRTG-2379,HE5386/18-1,19-2,22-1,23-1under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612+1 种基金The work of A.Kurganov was supported in part by the NSFC grant 12171226the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design,China(No.2019B030301001).
文摘In this paper,we develop new high-order numerical methods for hyperbolic systems of nonlinear partial differential equations(PDEs)with uncertainties.The new approach is realized in the semi-discrete finite-volume framework and is based on fifth-order weighted essentially non-oscillatory(WENO)interpolations in(multidimensional)random space combined with second-order piecewise linear reconstruction in physical space.Compared with spectral approximations in the random space,the presented methods are essentially non-oscillatory as they do not suffer from the Gibbs phenomenon while still achieving high-order accuracy.The new methods are tested on a number of numerical examples for both the Euler equations of gas dynamics and the Saint-Venant system of shallow-water equations.In the latter case,the methods are also proven to be well-balanced and positivity-preserving.
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 Key Basic Research and Development Program (No.2014CB744100)
文摘A novel third-order optimized symmetric weighted essentially non-oscillatory(WENO-OS3)scheme is used to simulate the hypersonic shock wave/boundary layer interactions.Firstly,the scheme is presented with the achievement of low dissipation in smooth region and robust shock-capturing capabilities in discontinuities.The Maxwell slip boundary conditions are employed to consider the rarefied effect near the surface.Secondly,several validating tests are given to show the good resolution of the WENO-OS3 scheme and the feasibility of the Maxwell slip boundary conditions.Finally,hypersonic flows around the hollow cylinder truncated flare(HCTF)and the25°/55°sharp double cone are studied.Discussions are made on the characteristics of the hypersonic shock wave/boundary layer interactions with and without the consideration of the slip effect.The results indicate that the scheme has a good capability in predicting heat transfer with a high resolution for describing fluid structures.With the slip boundary conditions,the separation region at the corner is smaller and the prediction is more accurate than that with no-slip boundary conditions.
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.
