The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity proper...The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.展开更多
In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme ...In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes.We relax the tangential continuity for velocity,which is the key ingredi-ent in achieving the uniform robustness.We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme,and the theories indicate that the error estimates for velocity are independent of pressure.Several numerical experiments are presented to confirm the theoretical findings.展开更多
For the numerical simulation of compressible flows,normally different mesh sizes are expected in different regions.For example,smaller mesh sizes are required to improve the local numerical resolution in the regions w...For the numerical simulation of compressible flows,normally different mesh sizes are expected in different regions.For example,smaller mesh sizes are required to improve the local numerical resolution in the regions where the physical variables vary violently(for example,near the shock waves or in the boundary layers)and larger elements are expected for the regions where the solution is smooth.h-adaptive mesh has been widely used for complex flows.However,there are two difficulties when employing h-adaptivity for high-order discontinuous Galerkin(DG)methods.First,locally curved elements are required to precisely match the solid boundary,which significantly increases the difficulty to conduct the"refining"and"coarsening"operations since the curved information has to be maintained.Second,h-adaptivity could break the partition balancing,which would significantly affect the efficiency of parallel computing.In this paper,a robust and automatic h-adaptive method is developed for high-order DG methods on locally curved tetrahedral mesh,for which the curved geometries are maintained during the h-adaptivity.Furthermore,the reallocating and rebalancing of the computational loads on parallel clusters are conducted to maintain the parallel efficiency.Numerical results indicate that the introduced h-adaptive method is able to generate more reasonable mesh according to the structure of flow-fields.展开更多
We consider the mixed discontinuous Galerkin(DG)finite element approximation of the Stokes equation and provide the analysis for the[P_(k)]^d-P_(k-1)element on the tensor product meshes.Comparing to the previous stabi...We consider the mixed discontinuous Galerkin(DG)finite element approximation of the Stokes equation and provide the analysis for the[P_(k)]^d-P_(k-1)element on the tensor product meshes.Comparing to the previous stability proof with[Q_(k)]^(d)-Q_(k-1)discontinuous finite elements in the existing references,our first contribution is to extend the formal proof to the[P_(k)]^d-P_(k-1)discontinuous elements on the tensor product meshes.Numerical infsup tests have been performed to compare Q_(x)and P_(k)types of elements and validate the well-posedness in both settings.Secondly,our contribution is to design the enhanced pressure-robust discretization by only modifying the body source assembling on[P_(k)]^d-P_(k-1)schemes to improve the numerical simulation further.The produced numerical velocity solution via our enhancement shows viscosity and pressure independence and thus outperforms the solution produced by standard discontinuous Galerkin schemes.Robustness analysis and numerical tests have been provided to validate the scheme's robustness.展开更多
A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability o...A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.展开更多
An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our sc...An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.展开更多
In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability...In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.展开更多
为解决有载调压变压器分接头(on-load tap changer,OLTC)、电容器组和分布式电源(distributed generation,DG)的协调优化问题,提出一种配电网动态无功优化方法。该方法采用最优分割法分别对OLTC和电容器组的静态最优投切序列进行有序聚...为解决有载调压变压器分接头(on-load tap changer,OLTC)、电容器组和分布式电源(distributed generation,DG)的协调优化问题,提出一种配电网动态无功优化方法。该方法采用最优分割法分别对OLTC和电容器组的静态最优投切序列进行有序聚类,在满足最大动作次数约束的前提下实现控制设备在时间上的解耦。分析了常见DG无功出力极限的影响因素,考虑OLTC、电容器组和DG在电压无功调节中的控制能力,提出三者的协调优化方法。由于OLTC的档位调节直接影响整个线路的电压无功分布,首先根据各时刻的静态优化结果和最优分割法确定OLTC的动作时刻及档位;然后,采用最优分割法确定电容器组的动作时刻,并将电容器组的投切容量和DG无功出力联合优化,得到最终的控制方案。最后,通过某实际电网算例验证了所提方法的合理性和有效性。展开更多
基金supported by the National Natural Science Foundation of China(12001189)supported by the National Natural Science Foundation of China(11171104,12171148)。
文摘The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.
基金the Hong Kong RGC General Research Fund(Project numbers 14304719 and 14302018)CUHK Faculty of Science Direct Grant 2019-20。
文摘In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes.We relax the tangential continuity for velocity,which is the key ingredi-ent in achieving the uniform robustness.We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme,and the theories indicate that the error estimates for velocity are independent of pressure.Several numerical experiments are presented to confirm the theoretical findings.
基金supported by the funding of the Key Laboratory of Aerodynamic Noise Control(No.ANCL20190103)the State Key Laboratory of Aerodynamics(No.SKLA20180102)+1 种基金the Aeronautical Science Foundation of China(Nos.2018ZA52002,2019ZA052011)the National Natural Science Foundation of China(Nos.61672281,61732006)。
文摘For the numerical simulation of compressible flows,normally different mesh sizes are expected in different regions.For example,smaller mesh sizes are required to improve the local numerical resolution in the regions where the physical variables vary violently(for example,near the shock waves or in the boundary layers)and larger elements are expected for the regions where the solution is smooth.h-adaptive mesh has been widely used for complex flows.However,there are two difficulties when employing h-adaptivity for high-order discontinuous Galerkin(DG)methods.First,locally curved elements are required to precisely match the solid boundary,which significantly increases the difficulty to conduct the"refining"and"coarsening"operations since the curved information has to be maintained.Second,h-adaptivity could break the partition balancing,which would significantly affect the efficiency of parallel computing.In this paper,a robust and automatic h-adaptive method is developed for high-order DG methods on locally curved tetrahedral mesh,for which the curved geometries are maintained during the h-adaptivity.Furthermore,the reallocating and rebalancing of the computational loads on parallel clusters are conducted to maintain the parallel efficiency.Numerical results indicate that the introduced h-adaptive method is able to generate more reasonable mesh according to the structure of flow-fields.
文摘We consider the mixed discontinuous Galerkin(DG)finite element approximation of the Stokes equation and provide the analysis for the[P_(k)]^d-P_(k-1)element on the tensor product meshes.Comparing to the previous stability proof with[Q_(k)]^(d)-Q_(k-1)discontinuous finite elements in the existing references,our first contribution is to extend the formal proof to the[P_(k)]^d-P_(k-1)discontinuous elements on the tensor product meshes.Numerical infsup tests have been performed to compare Q_(x)and P_(k)types of elements and validate the well-posedness in both settings.Secondly,our contribution is to design the enhanced pressure-robust discretization by only modifying the body source assembling on[P_(k)]^d-P_(k-1)schemes to improve the numerical simulation further.The produced numerical velocity solution via our enhancement shows viscosity and pressure independence and thus outperforms the solution produced by standard discontinuous Galerkin schemes.Robustness analysis and numerical tests have been provided to validate the scheme's robustness.
基金J.Qiu is supported partly by National Natural Science Foundation(China)grant 12071392.
文摘A high-order, well-balanced, positivity-preserving quasi-Lagrange movingmesh DG method is presented for the shallow water equations with non-flat bottomtopography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake ortsunami waves in the deep ocean. The method combines a quasi-Lagrange movingmesh DG method, a hydrostatic reconstruction technique, and a change of unknownvariables. The strategies in the use of slope limiting, positivity-preservation limiting,and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treatsmesh movement continuously in time and has the advantages that it does not need tointerpolate flow variables from the old mesh to the new one and places no constraintfor the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the wellbalance property, positivity preservation, and high-order accuracy of the method andits ability to adapt the mesh according to features in the flow and bottom topography.
基金supported by the National Natural Science Foundation of China(Grant Nos.11171104,91430107)the Construct Program of the Key Discipline in Hunan.This first author is supported by Hunan Provincial Innovation Foundation for Postgraduate under Grant CX2013B217.
文摘An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.
基金supported by NSFC(11341002)NSFC(11171104,10871066)+1 种基金the Construct Program of the Key Discipline in Hunansupported in part by US National Science Foundation under Grant DMS-1115530
文摘In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.
文摘为解决有载调压变压器分接头(on-load tap changer,OLTC)、电容器组和分布式电源(distributed generation,DG)的协调优化问题,提出一种配电网动态无功优化方法。该方法采用最优分割法分别对OLTC和电容器组的静态最优投切序列进行有序聚类,在满足最大动作次数约束的前提下实现控制设备在时间上的解耦。分析了常见DG无功出力极限的影响因素,考虑OLTC、电容器组和DG在电压无功调节中的控制能力,提出三者的协调优化方法。由于OLTC的档位调节直接影响整个线路的电压无功分布,首先根据各时刻的静态优化结果和最优分割法确定OLTC的动作时刻及档位;然后,采用最优分割法确定电容器组的动作时刻,并将电容器组的投切容量和DG无功出力联合优化,得到最终的控制方案。最后,通过某实际电网算例验证了所提方法的合理性和有效性。
