A set of small-stencil new Pade schemes with the same denominator are presented to solve high-order nonlinear evolution equations. Using this scheme, the fourth-order precision can not only be kept, but also the final...A set of small-stencil new Pade schemes with the same denominator are presented to solve high-order nonlinear evolution equations. Using this scheme, the fourth-order precision can not only be kept, but also the final three-diagonal discrete systems are solved by simple Doolittle methods, or ODE systems by Runge-Kutta technique. Numerical samples show that the schemes are very satisfactory. And the advantage of the schemes is very clear compared to other finite difference schemes.展开更多
The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-...The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.展开更多
We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations.High accuracy(up to the sixth-order ...We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations.High accuracy(up to the sixth-order presently)is achieved,thanks to polynomial recon-structions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of dis-continuities.We supplemented this scheme with a stencil construction allowing to reduce even further the numerical dissipation.The stencil is shifted away from troubles(shocks,discontinuities,etc.)leading to less oscillating polynomial reconstructions.Experimented on linear,Burgers',and Euler equations,we demonstrate that the adaptive stencil technique manages to retrieve smooth solutions with optimal order of accuracy but also irregular ones without spurious oscillations.Moreover,we numerically show that the approach allows to reduce the dissipation still maintaining the essentially non-oscillatory behavior.展开更多
Improved Weighted Essentially Non-oscillatory Scheme is a high order finite volume method. The mixed stencils can be obtained by a combination of r + 1 order and r order stencils. We improve the weights by the mapping...Improved Weighted Essentially Non-oscillatory Scheme is a high order finite volume method. The mixed stencils can be obtained by a combination of r + 1 order and r order stencils. We improve the weights by the mapping method. The restriction that conventional ENO or WENO schemes only use r order stencils, is removed. Higher resolution can be achieved by introducing the r + 1 order stencils. This method is verified by three cases, i.e. the interaction of a moving shock with a density wave problem, the interacting blast wave problem and the double mach reflection problem. The numerical results show that the Improved Weighted Essential Non-oscillatory method is a stable, accurate high-resolution finite volume scheme.展开更多
In a computing platform composed of several homogeneous processors,any parallel schedule of an algorithm usually involves three basic costs:arithmetic throughput on each processor,data movement between processors,and ...In a computing platform composed of several homogeneous processors,any parallel schedule of an algorithm usually involves three basic costs:arithmetic throughput on each processor,data movement between processors,and synchronization latency for several processors.The trade-offs between these three costs could realistically reflect lower bounds on the execution time for an algorithm.Therefore,the trade-off analysis is important for evaluating the optimality of a proposed schedule,and often yields new insights in parallel optimization.In this paper,we focus on the trade-offs between computation,communication,and synchronization in the stencil-collective alternate update,which is often executed repeatedly by the complex workflow with multiple stages in most numerical methods,such as the conjugate gradient(CG)method,the nonlinear time integration method in the dynamical core of a global atmospheric general circulation model(AGCM),and so on.Firstly,in order to formalize a workflow with multiple different stages,a novel operator representation of parallel algorithms is proposed.Based on the operator representation,we find the minimum vertex separator of the dependency graph for a stencil-collective alternate update.This breakthrough brings us the opportunity to obtain the cost lower bounds.Next,the general trade-off theory of the stencil-collective alternate update is founded successfully,which extends the recent trade-off theory to a more general theoretical context.Finally,by applying the general theoretical result to several algorithms,namely CG method and the nonlinear time integration method in AGCM,we obtain the corresponding lower bounds of computational cost,communication throughput,and synchronization latency.It should be noted that the general theory can also be widely used to analyze other complex numerical methods in real-world applications.展开更多
文摘A set of small-stencil new Pade schemes with the same denominator are presented to solve high-order nonlinear evolution equations. Using this scheme, the fourth-order precision can not only be kept, but also the final three-diagonal discrete systems are solved by simple Doolittle methods, or ODE systems by Runge-Kutta technique. Numerical samples show that the schemes are very satisfactory. And the advantage of the schemes is very clear compared to other finite difference schemes.
基金Project supported by the National Key Project, China (Grant No. GJXM92579).
文摘The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.
基金support by FEDER-Fundo Europeu de Desenvolvimento Regional,through COMPETE 2020-Programa Operational Fatores de Competitividade,and the National Funds through FCT-Fundacao para a Ciencia e a Tecnologia,project no.UID/FIS/04650/2019support by FEDER-Fundo Europeu de Desenvolvimento Regional,through COMPETI E 2020-Programa Operacional Fatores de Competitividade,and the National Funds through FCT-Fundacao para a Ciencia e a Tecnologia,project no.POCI-01-0145-FEDER-028118
文摘We propose an adaptive stencil construction for high-order accurate finite volume schemes a posteriori stabilized devoted to solve one-dimensional steady-state hyperbolic equations.High accuracy(up to the sixth-order presently)is achieved,thanks to polynomial recon-structions while stability is provided with an a posteriori MOOD method which controls the cell polynomial degree for eliminating non-physical oscillations in the vicinity of dis-continuities.We supplemented this scheme with a stencil construction allowing to reduce even further the numerical dissipation.The stencil is shifted away from troubles(shocks,discontinuities,etc.)leading to less oscillating polynomial reconstructions.Experimented on linear,Burgers',and Euler equations,we demonstrate that the adaptive stencil technique manages to retrieve smooth solutions with optimal order of accuracy but also irregular ones without spurious oscillations.Moreover,we numerically show that the approach allows to reduce the dissipation still maintaining the essentially non-oscillatory behavior.
文摘Improved Weighted Essentially Non-oscillatory Scheme is a high order finite volume method. The mixed stencils can be obtained by a combination of r + 1 order and r order stencils. We improve the weights by the mapping method. The restriction that conventional ENO or WENO schemes only use r order stencils, is removed. Higher resolution can be achieved by introducing the r + 1 order stencils. This method is verified by three cases, i.e. the interaction of a moving shock with a density wave problem, the interacting blast wave problem and the double mach reflection problem. The numerical results show that the Improved Weighted Essential Non-oscillatory method is a stable, accurate high-resolution finite volume scheme.
基金supported by the National Key Research and Development Program of China under Grant No.2016YFB0200800National Natural Science Foundation of China under Grant No.61802369.
文摘In a computing platform composed of several homogeneous processors,any parallel schedule of an algorithm usually involves three basic costs:arithmetic throughput on each processor,data movement between processors,and synchronization latency for several processors.The trade-offs between these three costs could realistically reflect lower bounds on the execution time for an algorithm.Therefore,the trade-off analysis is important for evaluating the optimality of a proposed schedule,and often yields new insights in parallel optimization.In this paper,we focus on the trade-offs between computation,communication,and synchronization in the stencil-collective alternate update,which is often executed repeatedly by the complex workflow with multiple stages in most numerical methods,such as the conjugate gradient(CG)method,the nonlinear time integration method in the dynamical core of a global atmospheric general circulation model(AGCM),and so on.Firstly,in order to formalize a workflow with multiple different stages,a novel operator representation of parallel algorithms is proposed.Based on the operator representation,we find the minimum vertex separator of the dependency graph for a stencil-collective alternate update.This breakthrough brings us the opportunity to obtain the cost lower bounds.Next,the general trade-off theory of the stencil-collective alternate update is founded successfully,which extends the recent trade-off theory to a more general theoretical context.Finally,by applying the general theoretical result to several algorithms,namely CG method and the nonlinear time integration method in AGCM,we obtain the corresponding lower bounds of computational cost,communication throughput,and synchronization latency.It should be noted that the general theory can also be widely used to analyze other complex numerical methods in real-world applications.
