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
In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbol...In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation laws.For the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more wider.The more we use wider candidate stencils,the bigger the probability of discontinuities lies in all stencils.Therefore,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils.By the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(J.Comput.Phys.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth regions.These new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL number.Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.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 present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the targe...In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.展开更多
Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternati...Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.展开更多
A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The e...A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.展开更多
In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridizat...In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridization,the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region;meanwhile,the essentially oscillation-free solution could also be obtained.By adapting the original smoothness indicator in the WENO reconstruction,the stencil is distinguished into three types:smooth,non-smooth,and high-frequency region.In the smooth region,the linear reconstruction is used and the non-smooth region with the WENO reconstruction.In the high-frequency region,the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor,which could capture high-frequency wave efficiently.Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.展开更多
We develop and use a novel mixed-precision weighted essentially non-oscillatory(WENO)method for solving the Teukolsky equation,which arises when modeling perturbations of Kerr black holes.We show that WENO methods out...We develop and use a novel mixed-precision weighted essentially non-oscillatory(WENO)method for solving the Teukolsky equation,which arises when modeling perturbations of Kerr black holes.We show that WENO methods outperform higher-order finite-difference methods,standard in the discretization of the Teukolsky equation,due to the need to add dissipation for stability purposes in the latter.In particular,as the WENO scheme uses no additional dissipation,it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio bina-ries.In the mixed-precision approach,the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of≈3.3.In addition,we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations.Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics.We apply our solver to study the behavior of the Aretakis charge—a conserved quantity,that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.展开更多
In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated...In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.展开更多
基金the National Natural Science Foundation of China(12071112)and(11471102)the Basic Research Projects for Key Scientific Research Projects in Henan Province(20ZX001)the Research and Practice Project on Education and Teaching Reform in Henan Institute of Science and Technology(2021YB45)。
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 NSFC grant 11872210 and the Science Challenge Project,No.TZ2016002the NSFC Grant 11926103 when he visited Tianyuan Mathematical Center in Southeast China,Xiamen 361005,Fujian,Chinathe NSFC Grant 12071392 and the Science Challenge Project,No.TZ2016002.
文摘In this paper,a new type of finite difference mapped weighted essentially non-oscillatory(MWENO)schemes with unequal-sized stencils,such as the seventh-order and ninthorder versions,is constructed for solving hyperbolic conservation laws.For the purpose of designing increasingly high-order finite difference WENO schemes,the equal-sized stencils are becoming more and more wider.The more we use wider candidate stencils,the bigger the probability of discontinuities lies in all stencils.Therefore,one innovation of these new WENO schemes is to introduce a new splitting stencil methodology to divide some fourpoint or five-point stencils into several smaller three-point stencils.By the usage of this new methodology in high-order spatial reconstruction procedure,we get different degree polynomials defined on these unequal-sized stencils,and calculate the linear weights,smoothness indicators,and nonlinear weights as specified in Jiang and Shu(J.Comput.Phys.126:202228,1996).Since the difference between the nonlinear weights and the linear weights is too big to keep the optimal order of accuracy in smooth regions,another crucial innovation is to present the new mapping functions which are used to obtain the mapped nonlinear weights and decrease the difference quantity between the mapped nonlinear weights and the linear weights,so as to keep the optimal order of accuracy in smooth regions.These new MWENO schemes can also be applied to compute some extreme examples,such as the double rarefaction wave problem,the Sedov blast wave problem,and the Leblanc problem with a normal CFL number.Extensive numerical results are provided to illustrate the good performance of the new finite difference MWENO schemes.
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AFOSR and NSF for their support of this work under grants FA9550-19-1-0281 and FA9550-17-1-0394 and NSF grant DMS 191218。
文摘In this paper,we present a semi-Lagrangian(SL)method based on a non-polynomial function space for solving the Vlasov equation.We fnd that a non-polynomial function based scheme is suitable to the specifcs of the target problems.To address issues that arise in phase space models of plasma problems,we develop a weighted essentially non-oscillatory(WENO)scheme using trigonometric polynomials.In particular,the non-polynomial WENO method is able to achieve improved accuracy near sharp gradients or discontinuities.Moreover,to obtain a high-order of accuracy in not only space but also time,it is proposed to apply a high-order splitting scheme in time.We aim to introduce the entire SL algorithm with high-order splitting in time and high-order WENO reconstruction in space to solve the Vlasov-Poisson system.Some numerical experiments are presented to demonstrate robustness of the proposed method in having a high-order of convergence and in capturing non-smooth solutions.A key observation is that the method can capture phase structure that require twice the resolution with a polynomial based method.In 6D,this would represent a signifcant savings.
基金Research was supported by the NSFC Grant 11872210Research was supported by the NSFC Grant 11872210 and Grant No.MCMS-I-0120G01+1 种基金Research supported in part by the AFOSR Grant FA9550-20-1-0055NSF Grant DMS-2010107.
文摘Fixed-point fast sweeping WENO methods are a class of efficient high-order numerical methods to solve steady-state solutions of hyperbolic partial differential equations(PDEs).The Gauss-Seidel iterations and alternating sweeping strategy are used to cover characteristics of hyperbolic PDEs in each sweeping order to achieve fast convergence rate to steady-state solutions.A nice property of fixed-point fast sweeping WENO methods which distinguishes them from other fast sweeping methods is that they are explicit and do not require inverse operation of nonlinear local systems.Hence,they are easy to be applied to a general hyperbolic system.To deal with the difficulties associated with numerical boundary treatment when high-order finite difference methods on a Cartesian mesh are used to solve hyperbolic PDEs on complex domains,inverse Lax-Wendroff(ILW)procedures were developed as a very effective approach in the literature.In this paper,we combine a fifthorder fixed-point fast sweeping WENO method with an ILW procedure to solve steadystate solution of hyperbolic conservation laws on complex computing regions.Numerical experiments are performed to test the method in solving various problems including the cases with the physical boundary not aligned with the grids.Numerical results show highorder accuracy and good performance of the method.Furthermore,the method is compared with the popular third-order total variation diminishing Runge-Kutta(TVD-RK3)time-marching method for steady-state computations.Numerical examples show that for most of examples,the fixed-point fast sweeping method saves more than half CPU time costs than TVD-RK3 to converge to steady-state solutions.
基金the National Natural Science Foundation of China(11901555,11871448,12001009).
文摘A high order finite difference numerical scheme is developed for the shallow water equations on curvilinear meshes based on an alternative flux formulation of the weighted essentially non-oscillatory(WENO)scheme.The exact C-property is investigated,and comparison with the standard finite difference WENO scheme is made.Theoretical derivation and numerical results show that the proposed finite difference WENO scheme can maintain the exact C-property on both stationarily and dynamically generalized coordinate systems.The Harten-Lax-van Leer type flux is developed on general curvilinear meshes in two dimensions and verified on a number of benchmark problems,indicating smaller errors compared with the Lax-Friedrichs solver.In addition,we propose a positivity-preserving limiter on stationary meshes such that the scheme can preserve the non-negativity of the water height without loss of mass conservation.
基金the National Numerical Windtunnel Project NNW2019ZT4-B08the NSFC grant No.11871449.
文摘In this paper,a new kind of hybrid method based on the weighted essentially non-oscillatory(WENO)type reconstruction is proposed to solve hyperbolic conservation laws.Comparing the WENO schemes with/without hybridization,the hybrid one can resolve more details in the region containing multi-scale structures and achieve higher resolution in the smooth region;meanwhile,the essentially oscillation-free solution could also be obtained.By adapting the original smoothness indicator in the WENO reconstruction,the stencil is distinguished into three types:smooth,non-smooth,and high-frequency region.In the smooth region,the linear reconstruction is used and the non-smooth region with the WENO reconstruction.In the high-frequency region,the mixed scheme of the linear and WENO schemes is adopted with the smoothness amplification factor,which could capture high-frequency wave efficiently.Spectral analysis and numerous examples are presented to demonstrate the robustness and performance of the hybrid scheme for hyperbolic conservation laws.
基金support of NSF Grants No.PHY-2010685(G.K)and No.DMS-1912716(S.F,S.G,and G.K),AFOSR Grant No.FA9550-18-1-0383(S.G)Office of Naval Research/Defense University Research Instrumentation Program(ONR/DURIP)Grant No.N00014181255+2 种基金the National Science Foundation under Grant No.DMS-1439786 while a subset of the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence,RI,during the Advances in Computational Relativity program.A part of this research is sponsored by the Office of Advanced Scientific Computing ResearchUS Department of Energy,and was performed at the Oak Ridge National Laboratory,which is managed by UT-Battelle,LLC under Contract no.De-AC05-00OR22725UT-Battelle,LLC,under contract DE-AC05-00OR22725 with the US Department of Energy.
文摘We develop and use a novel mixed-precision weighted essentially non-oscillatory(WENO)method for solving the Teukolsky equation,which arises when modeling perturbations of Kerr black holes.We show that WENO methods outperform higher-order finite-difference methods,standard in the discretization of the Teukolsky equation,due to the need to add dissipation for stability purposes in the latter.In particular,as the WENO scheme uses no additional dissipation,it is well suited for scenarios requiring long-time evolution such as the study of price tails and gravitational wave emission from extreme mass ratio bina-ries.In the mixed-precision approach,the expensive computation of the WENO weights is performed in reduced floating-point precision that results in a significant speedup factor of≈3.3.In addition,we use state-of-the-art Nvidia general-purpose graphics processing units and cluster parallelism to further accelerate the WENO computations.Our optimized WENO solver can be used to quickly generate accurate results of significance in the field of black hole and gravitational wave physics.We apply our solver to study the behavior of the Aretakis charge—a conserved quantity,that if detected by a gravitational wave observatory like LIGO/Virgo would prove the existence of extremal black holes.
基金the National Natural Science Foundation of China(91641107,91852116,12071470)Fundamental Research of Civil Aircraft(MJ-F-2012-04)of Ministry of Industrialization and Information of China.
文摘In this paper,we develop a new sixth-order WENO scheme by adopting a convex combina-tion of a sixth-order global reconstruction and four low-order local reconstructions.Unlike the classical WENO schemes,the associated linear weights of the new scheme can be any positive numbers with the only requirement that their sum equals one.Further,a very simple smoothness indicator for the global stencil is proposed.The new scheme can achieve sixth-order accuracy in smooth regions.Numerical tests in some one-and two-dimensional bench-mark problems show that the new scheme has a little bit higher resolution compared with the recently developed sixth-order WENO-Z6 scheme,and it is more efficient than the classical fifth-order WENO-JS5 scheme and the recently developed sixth-order WENO6-S scheme.
