Mimetic finite difference method

Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail

doi:10.1016/j.jcp.2013.07.031

Cited by 394 publications

(275 citation statements)

References 173 publications

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“…Originally, the FDTD was restricted to the structured grids constructed by cubic elements, but there are certain generalizations to, e.g., three-dimensional nonuniform [39] and non-orthogonal [32] grids and two-dimensional polyhedra [27]. Other physics-preserving methods that work in the style of staggered finite difference grids include, e.g., mimetic finite difference (MFD) schemes [34].…”

mentioning

confidence: 99%

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

Räbinä¹,

Mönkölä²,

Rossi³

2015

SIAM J. Sci. Comput.

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Abstract. We consider the computationally efficient time integration of Maxwell's equations using discrete exterior calculus (DEC) as the computational framework. With the theory of DEC, we associate the degrees of freedom of the electric and magnetic fields with primal and dual mesh structures, respectively. We concentrate on mesh constructions that imitate the geometry of the close packing in crystal lattices that is typical of elemental metals and intermetallic compounds. This class of computational grids has not been used previously in electromagnetics. For the simulation of wave propagation driven by time-harmonic source terms, we provide an optimized Hodge operator and a novel time discretization scheme with nonuniform time step size. The numerical experiments show a significant improvement in accuracy and a decrease in computing time compared to simulations with well-known variants of the finite difference time domain method.

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confidence: 99%

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

Räbinä¹,

Mönkölä²,

Rossi³

2015

SIAM J. Sci. Comput.

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“…Additional vorticity degrees of freedom are located at the vertices of the Voronoi grid cells. Such an arrangement facilitates the construction of a standard conservative finite-volume-type scheme for the transport of cell-centred fluid properties, and a mimeticclass (Lipnikov et al, 2014;Bochev and Hyman, 2006) finite-difference formulation for the evolution of velocity components. Overall, this scheme is known to posses a variety of desirable conservation properties, conserving mass, potential vorticity and enstrophy, and preserving geostrophic balance (Ringler et al, 2010).…”

Section: Grid Generation For General Circulation Modellingmentioning

confidence: 99%

JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere

Engwirda

2017

Geosci. Model Dev.

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Abstract. An algorithm for the generation of non-uniform, locally orthogonal staggered unstructured spheroidal grids is described. This technique is designed to generate very high-quality staggered Voronoi-Delaunay meshes appropriate for general circulation modelling on the sphere, including applications to atmospheric simulation, ocean-modelling and numerical weather prediction. Using a recently developed Frontal-Delaunay refinement technique, a method for the construction of high-quality unstructured spheroidal Delaunay triangulations is introduced. A locally orthogonal polygonal grid, derived from the associated Voronoi diagram, is computed as the staggered dual. It is shown that use of the Frontal-Delaunay refinement technique allows for the generation of very high-quality unstructured triangulations, satisfying a priori bounds on element size and shape. Grid quality is further improved through the application of hill-climbing-type optimisation techniques. Overall, the algorithm is shown to produce grids with very high element quality and smooth grading characteristics, while imposing relatively low computational expense. A selection of uniform and non-uniform spheroidal grids appropriate for highresolution, multi-scale general circulation modelling are presented. These grids are shown to satisfy the geometric constraints associated with contemporary unstructured C-gridtype finite-volume models, including the Model for Prediction Across Scales (MPAS-O). The use of user-defined meshspacing functions to generate smoothly graded, non-uniform grids for multi-resolution-type studies is discussed in detail.

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“…Among the members of this family, we can cite the covolume method [117], spline-based cochain discretization [118], mimetic spectral elements [119][120][121] or spectral DEC [122,123]. Other compatible discretization techniques include the Finite Element Exterior Calculus (FEEC) [124][125][126], mimetic finite differences ( [127] and references therein) and works in [128][129][130][131][132].…”

Section: Introductionmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Mimetic finite difference method

Cited by 394 publications

References 173 publications

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

Efficient Time Integration of Maxwell's Equations with Generalized Finite Differences

JIGSAW-GEO (1.0): locally orthogonal staggered unstructured grid generation for general circulation modelling on the sphere

A review of some geometric integrators

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