A method of calculating the safety profile on the HT-7 tokamak has been described in this paper. It is derived from Maxwell's equations, among which we-mainly use .two of them: one is the magnetic field diffusion ...A method of calculating the safety profile on the HT-7 tokamak has been described in this paper. It is derived from Maxwell's equations, among which we-mainly use .two of them: one is the magnetic field diffusion equation, and the other is Ampere's Law. This method can be also used to evaluate the safety factor on other devices with a circular cross sections. It is helpful to the study of the plasma MHD behavior on the HT-7 tokamak.展开更多
In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system....In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultaneously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(i)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ii)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a corresponding conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.展开更多
The extended Hermite interpolation problem on segment points set over n-dimensional Euclidean space is considered. Based on the algorithm to compute the Gr?bner basis of Ideal given by dual basis a new method to const...The extended Hermite interpolation problem on segment points set over n-dimensional Euclidean space is considered. Based on the algorithm to compute the Gr?bner basis of Ideal given by dual basis a new method to construct minimal multivariate polynomial which satisfies the interpolation conditions is given.展开更多
文摘A method of calculating the safety profile on the HT-7 tokamak has been described in this paper. It is derived from Maxwell's equations, among which we-mainly use .two of them: one is the magnetic field diffusion equation, and the other is Ampere's Law. This method can be also used to evaluate the safety factor on other devices with a circular cross sections. It is helpful to the study of the plasma MHD behavior on the HT-7 tokamak.
基金supported by the NSF(Grant Nos.the NSF-DMS-1818924 and 2111253)the Air Force Office of Scientific Research FA9550-22-1-0390 and Department of Energy DE-SC0023164+1 种基金supported by the NSF(Grant Nos.NSF-DMS-1830838 and NSF-DMS-2111383)the Air Force Office of Scientific Research FA9550-22-1-0390.
文摘In this paper,we propose a novel Local Macroscopic Conservative(LoMaC)low rank tensor method with discontinuous Galerkin(DG)discretization for the physical and phase spaces for simulating the Vlasov-Poisson(VP)system.The LoMaC property refers to the exact local conservation of macroscopic mass,momentum,and energy at the discrete level.The recently developed LoMaC low rank tensor algorithm(arXiv:2207.00518)simultaneously evolves the macroscopic conservation laws of mass,momentum,and energy using the kinetic flux vector splitting;then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables.This paper is a generalization of our previous work,but with DG discretization to take advantage of its compactness and flexibility in handling boundary conditions and its superior accuracy in the long term.The algorithm is developed in a similar fashion as that for a finite difference scheme,by observing that the DG method can be viewed equivalently in a nodal fashion.With the nodal DG method,assuming a tensorized computational grid,one will be able to(i)derive differentiation matrices for different nodal points based on a DG upwind discretization of transport terms,and(ii)define a weighted inner product space based on the nodal DG grid points.The algorithm can be extended to the high dimensional problems by hierarchical Tucker(HT)decomposition of solution tensors and a corresponding conservative projection algorithm.In a similar spirit,the algorithm can be extended to DG methods on nodal points of an unstructured mesh,or to other types of discretization,e.g.,the spectral method in velocity direction.Extensive numerical results are performed to showcase the efficacy of the method.
文摘The extended Hermite interpolation problem on segment points set over n-dimensional Euclidean space is considered. Based on the algorithm to compute the Gr?bner basis of Ideal given by dual basis a new method to construct minimal multivariate polynomial which satisfies the interpolation conditions is given.
