Causal-Path Local Time-Stepping in the discontinuous Galerkin method for Maxwellʼs equations

Angulo, Luis D.; Alvarez, Jesus; Teixeira, F.L.; Pantoja, Mario F.; Garcia, S.G.

doi:10.1016/j.jcp.2013.09.010

Cited by 19 publications

(37 citation statements)

References 28 publications

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“…Next, we approximate in (3.5) the values of y n+ ξ t = t n , by their Taylor expansions up to O(Δt 4 ). We also interpolate the points (t n , F n ), (t n+ 1 2 , F n+ 1 2 ), and (t n+1 , F n+1 ) by the quadratic polynomial,…”

Section: The Rk4-based Lts Methodmentioning

confidence: 99%

Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation

Grote¹,

Mehlin²,

Mitkova³

2015

SIAM J. Sci. Comput.

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Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability, and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.1. Introduction. The efficient numerical simulation of time-dependent wave phenomena is of fundamental importance in acoustic, electromagnetic, or seismic wave propagation. In the presence of heterogeneous media or complex geometry, finite element methods (FEM), be they continuous or discontinuous, are increasingly popular due to their inherent flexibility: elements can be small precisely where small features are located and larger elsewhere. Local mesh refinement, however, also imposes severe stability constraints on explicit time integration, as the maximal time-step is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small region, the use of implicit methods, or a very small time-step in the entire computational domain, are generally too high a price to pay. Local time-stepping (LTS) methods alleviate that geometry, induced stability restriction by dividing the elements into two distinct regions: the "coarse region," which contains the larger elements and is integrated in time using an explicit method, and the "fine region," which contains the smaller elements and is integrated in time using either smaller time-steps or an implicit scheme.Locally implicit methods build on the long tradition of hybrid implicit-explicit (IMEX) algorithms for operator splitting in computational fluid dynamics-see [36,2] and the references therein. In 2006, Piperno [37] combined the explicit leap-frog (LF) with the implicit Crank-Nicolson (CN) scheme for a nodal discontinuous Galerkin (DG) discretization of Maxwell's equations in a nonconducting medium. Here, a linear system needs to be solved inside the refined region at every time-step. Although each method is time accurate of order two, the implicit-explicit component splitting

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Section: The Rk4-based Lts Methodmentioning

confidence: 99%

Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation

Grote¹,

Mehlin²,

Mitkova³

2015

SIAM J. Sci. Comput.

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“…During the last decade, the discontinuous Galerkin (DG) finite element method, first proposed by Reed and Hill [1], has become very popular for the solution of various problems from the field of fluid mechanics [2][3][4], electrodynamics or electromagnetism [5][6][7] and plasma physics [8]. The reasons behind the popularity can be attributed to the method's ability to achieve high order of spatial accuracy combined with low artificial damping, robustness and overall stability, thus, making it an ideal method for the simulation of laminar and turbulent flows with complex vortical structures.…”

Section: Introductionmentioning

confidence: 99%

“…In this case, the smallest element can significantly reduce the size of the global time step and, thus, hamper the overall computational efficiency. To overcome the disadvantage, a local time-stepping (LTS) technique can be used, see, e.g., [7]. As the name suggests, the LTS method utilises the principle of local time steps that are computed for each element independently.…”

Section: Introductionmentioning

confidence: 99%

Comparison of discontinuous Galerkin time integration schemes for the solution of flow problems with deformable domains

Bublík

Vimmr

Jonášová

2015

Applied Mathematics and Computation

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“…leap-frog (LF2) [11]- [14] or the fourth-order low-storage explicit Runge-Kutta (LSERK4) [9], [15]- [18]. The maximum time step allowed for stability by these schemes is constrained by the spectral radius of the spatial operator, which in turns depend on the inverse square of the polynomial order and on the minimum edge length used for the spatial discretization.…”

mentioning

confidence: 99%

“…The use of -refinement in DGTD, also becomes problematic in multiscale problems, since (local) smaller elements enforce reduced (global) time-steps to ensure stability. Strategies to mitigate this exist, like local time-stepping techniques [12], [14], [15], and implicit-explicit (IMEX) time-integration schemes [22], [23]. Reduction on the spectral radius has been achieved using co-volume filtering [24] and mapping techniques [25], though these are effective only for higher orders.…”

mentioning

confidence: 99%

A Nodal Continuous-Discontinuous Galerkin Time-Domain Method for Maxwell's Equations

Angulo

Alvarez

Teixeira

et al. 2015

IEEE Trans. Microwave Theory Techn.

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A new nodal hybrid continuous-discontinuous Galerkin time-domain (CDGTD) method for the solution of Maxwell's curl equations is proposed and analyzed. This hybridization is made by clustering small collections of elements with a continuous Galerkin (CG) formalism. These clusters exchange information with their exterior through a discontinuous Galerkin (DG) numerical flux. This scheme shows reduced numerical dispersion error with respect to classical DG formulations for certain orders and numbers of clustered elements. The spectral radius of the clustered semi-discretized operator is smaller than its DG counterpart allowing for larger time steps in explicit time integrators. Additionally, the continuity across the element boundaries allows us a reduction of the number of degrees of freedom of up to about 80% for a low-order three-dimensional implementation. Index Terms-Continuous-discontinuous Galerkin time-domain (CDGTD), continuous Galerkin (CG) method, discontinuous Galerkin (DG) method, discontinuous Galerkin time-domain (DGTD), Maxwell's equations.

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Causal-Path Local Time-Stepping in the discontinuous Galerkin method for Maxwellʼs equations

Cited by 19 publications

References 28 publications

Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation

Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation

Comparison of discontinuous Galerkin time integration schemes for the solution of flow problems with deformable domains

A Nodal Continuous-Discontinuous Galerkin Time-Domain Method for Maxwell's Equations

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