A Time-Space Adaptive Method for the Schrödinger Equation

Kormann, Katharina

doi:10.4208/cicp.101214.021015a

Cited by 13 publications

(7 citation statements)

References 50 publications

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“…The concept of matrix-free evaluation with sum factorization has been widely adopted by now, like in the deal.II [1], DUNE [5,40,60], Firedrake [63], mfem [2], Nek5000 [28] or Nektar++ [13] projects. These fast evaluation techniques are directly applicable to explicit time stepping schemes, as we have demonstrated for wave propagation in [42,53,[65][66][67][68] and the compressible Navier-Stokes equations [24]. The proposed developments make matrix-free evaluation of highorder DG operators reach a throughput in unknowns per second almost as high as for optimized 5-wide finite difference stencils in a CFD context [75], despite delivering much higher accuracy.…”

Section: Implementation Of Sum Factorization In the Dealii Librarymentioning

confidence: 95%

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

Arndt

Fehn

Kanschat

et al. 2020

Lecture Notes in Computational Science and Engineering

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This text presents contributions to efficient high-order finite element solvers in the context of the project ExaDG, part of the DFG priority program 1648 Software for Exascale Computing (SPPEXA). The main algorithmic components are the matrix-free evaluation of finite element and discontinuous Galerkin operators with sum factorization to reach a high node-level performance and parallel scalability, a massively parallel multigrid framework, and efficient multigrid smoothers. The algorithms have been applied in a computational fluid dynamics context. The software contributions of the project have led to a speedup by a factor 3 − 4 depending on the hardware. Our implementations are available via the deal.II finite element library.

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Section: Implementation Of Sum Factorization In the Dealii Librarymentioning

confidence: 95%

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

Arndt

Fehn

Kanschat

et al. 2020

Lecture Notes in Computational Science and Engineering

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“…In case of an approximate solution that is defined continuously on the whole domain, the residual is defined as the difference between the continuous differential operator and the approximate difference operator applied to the approximate solution. This is possible for finite element approximations where a representation of the approximate solution based on some basis functions is known (see [9]). In our finite difference setting, we instead use a better approximation of the operator as a reference.…”

Section: Error Estimationmentioning

confidence: 99%

Stable Difference Methods for Block-Oriented Adaptive Grids

et al. 2014

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In this paper, we present a block-oriented scheme for adaptive mesh refinement based on summation-by-parts (SBP) finite difference methods and simultaneous-approximation-term (SAT) interface treatment. Since the order of accuracy at SBP-SAT grid interfaces is lower compared to that of the interior stencils, we strive at using the interior stencils across block-boundaries whenever possible. We devise a stable treatment of SBP-FD junction points, i.e. points where interfaces with different boundary treatment meet. This leads to stable discretizations for more flexible grid configurations within the SBP-SAT framework, with a reduced number of SBP-SAT interfaces. Both first and second derivatives are considered in the analysis. Even though the stencil order is locally reduced close to numerical interfaces and corner points, numerical simulations show that the locally reduced accuracy does not severely reduce the accuracy of the time propagated numerical solution. Moreover, we explain how to organize the grid and how to automatically adapt the mesh, aiming at problems of many variables. Examples of adaptive grids are demonstrated for the simulation of the time-dependent Schrödinger equation and for the advection equation.

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“…For the efficient implementation of the FEM model introduced in Sec. 2, we use a method based on [12] and [26]. To this end we choose (tensor) Gauss-Lobatto nodes of degree p on rectangular elements for the definition of the nodal basis and for the numerical evaluation of the inner products in (2.16).…”

Section: Implementation Aspectsmentioning

confidence: 99%

Convergence of a Strang splitting finite element discretization for the Schrödinger–Poisson equation

et al. 2017

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Abstract. Operator splitting methods combined with finite element spatial discretizations are studied for time-dependent nonlinear Schrödinger equations. In particular, the Schrödinger-Poisson equation under homogeneous Dirichlet boundary conditions on a finite domain is considered. A rigorous stability and error analysis is carried out for the second-order Strang splitting method and conforming polynomial finite element discretizations. For sufficiently regular solutions the classical orders of convergence are retained, that is, second-order convergence in time and polynomial convergence in space is proven. The established convergence result is confirmed and complemented by numerical illustrations.1991 Mathematics Subject Classification. 65J15, 65L05, 65M60 65M12, 65M15..

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A Time-Space Adaptive Method for the Schrödinger Equation

Cited by 13 publications

References 50 publications

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

Stable Difference Methods for Block-Oriented Adaptive Grids

Convergence of a Strang splitting finite element discretization for the Schrödinger–Poisson equation

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