Unconditionally Stable Time Stepping Method for Mixed Finite Element Maxwell Solvers

Crawford, Zane D.; Li, Jie; Christlieb, Andrew; Shanker, B.

doi:10.2528/pierc20021001

Cited by 15 publications

(11 citation statements)

References 9 publications

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“…The mixed finite element system in (2) is discretized in time using Newmark-Beta, an unconditionally stable time stepping method. This method has been extensively used in for the wave equation 11 and examined for the mixed finite element method in 17 , allowing for much larger time step sizes than the traditional leapfrog method. In this method, the fields in time are represented by three temporal basis functions…”

Section: B Unconditionally Stable Time Marchingmentioning

confidence: 99%

Time Integrator Agnostic Charge Conserving Finite Element PIC

O'Connor,

Crawford,

Ramachandran

et al. 2021

Preprint

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Developing particle-in-cell (PIC) methods using finite element basis sets, and without auxiliary divergence cleaning methods, was a long standing problem until recently. It was shown that if consistent spatial basis functions are used, one can indeed create a methodology that was charge conserving, albeit using a leap-frog time stepping method. While this is a significant advance, leap frog schemes are only conditionally stable and time step sizes are closely tied to the underlying mesh. Ideally, to take full advantage of advances in finite element methods (FEMs), one needs a charge conserving PIC methodology that is agnostic to the time stepping method. This is the principal contribution of this paper. In what follows, we shall develop this methodology, prove that both charge and Gauss' laws are discretely satisfied at every time step, provide the necessary details to implement this methodology for both the wave equation FEM and Maxwell Solver FEM, and finally demonstrate its efficacy on a suite of test problems. The method will be demonstrated by single particle evolution, non-neutral beams with space-charge, and adiabatic expansion of a neutral plasma, where the debye length has been resolved, and real mass ratios are used.

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Section: B Unconditionally Stable Time Marchingmentioning

confidence: 99%

Time Integrator Agnostic Charge Conserving Finite Element PIC

O'Connor,

Crawford,

Ramachandran

et al. 2021

Preprint

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“…To verify that the resulting system is unconditionally stable and examine its null space, we find the eigenvalues of the discrete system as described in Ref. 47 . Our discrete system is obtained from discretizing a cube of side length 1m using tetrahedra with average edge length of 0.214m.…”

Section: Eigenvaluesmentioning

confidence: 99%

“…Neumann boundary conditions are equivalent to a plane wave polarized along −ẑ and propagating along x direction, with temporal variation being the derivative of modulated Gaussian with center frequency of 10 MHz and bandwidth of 5 Mhz. The explicit formulae is fairly standard and can be obtained from say 47 .…”

Section: Neumann Boundary Conditionmentioning

confidence: 99%

Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell

O'Connor,

Crawford,

Ramachandran

et al. 2021

Preprint

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Development of particle in cell methods using finite element based methods (FEMs) have been a topic of renewed interest; this has largely been driven by (a) the ability of finite element methods to better model geometry, (b) better understanding of function spaces that are necessary to represent all Maxwell quantities, and (c) more recently, the fundamental rubrics that should be obeyed in space and time so as to satisfy Gauss' laws and the equation of continuity. In that vein, methods have been developed recently that satisfy these equations and are agnostic to time stepping methods. While is development is indeed a significant advance, it should be noted that implicit FEM transient solvers support an underlying null space that corresponds to a gradient of a scalar potential ∇Φ(r) (or t∇Φ(r) in the case of wave equation solvers). While explicit schemes do not suffer from this drawback, they are only conditionally stable, time step sizes are mesh dependent, and very small. A way to overcome this bottleneck, and indeed, satisfy all four Maxwell's equation is to use a quasi-Helmholtz formulation on a tesselation. In the re-formulation presented, we strictly satisfy the equation of continuity and Gauss' laws for both the electric and magnetic flux densities. Results demonstrating the efficacy of this scheme will be presented.

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“…The fields are evolved in time using an unconditionally stable Newmark-beta time marching scheme [28], [29]. This allows larger time steps than would be afforded by a leapfrog method.…”

Section: B Evolution Of Particle Path and Current Mappingmentioning

confidence: 99%

Higher Order Charge Conserving Electromagnetic Finite Element Particle in Cell Method

Crawford¹,

Ramachandran²,

O'Connor³

et al. 2021

Preprint

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Until recently, electromagnetic finite element PIC (EM-FEMPIC) methods that demonstrated charge conservation used explicit field solvers. It is only recently, that a series of papers developed the mathematics necessary for charge conservation within an implicit field solve and demonstrated for a number of examples. This permits using time steps sizes that are necessary to capture the physics as opposed to being restricted to those constrained by geometry. One aspect that is missing is higher order basis functions to represent both fields and particles. Higher order basis can be particularly helpful in effectively capturing complex field layouts with fewer degrees of freedom. Developing a framework for higher order EM-FEMPIC that maintains stability, improves accuracy, and conserves charge is the principal goal of this paper. A number of results are presented that attest to its efficacy.

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Unconditionally Stable Time Stepping Method for Mixed Finite Element Maxwell Solvers

Cited by 15 publications

References 9 publications

Time Integrator Agnostic Charge Conserving Finite Element PIC

Time Integrator Agnostic Charge Conserving Finite Element PIC

Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell

Higher Order Charge Conserving Electromagnetic Finite Element Particle in Cell Method

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