Modified Trigonometric Integrators

McLachlan, Robert I.; Stern, Ari

doi:10.1137/130921118

Cited by 27 publications

(51 citation statements)

References 18 publications

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“…For splitting/composition methods, it can sometimes be desirable to compute an approximate flow, even when the exact flow is computable. For example, the IMEX method for highlyoscillatory problems (Stern and Grinspun [21], McLachlan and Stern [18]) uses the midpoint method instead of the exact flow of a harmonic oscillator, and the Boris method for charged particle dynamics uses the midpoint method instead of the exact flow of the magnetic field (Hairer and Lubich [6, Eq. 1.4]).…”

Section: Numerical Integrators For Conditionally Linear Systemsmentioning

confidence: 99%

Structure-Preserving Numerical Integrators for Hodgkin--Huxley-Type Systems

Chen¹,

Raman²,

Stern³

2020

SIAM J. Sci. Comput.

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Motivated by the Hodgkin-Huxley model of neuronal dynamics, we study explicit numerical integrators for "conditionally linear" systems of ordinary differential equations. We show that splitting and composition methods, when applied to the Van der Pol oscillator and to the Hodgkin-Huxley model, do a better job of preserving limit cycles of these systems for large time steps, compared with the "Euler-type" methods (including Euler's method, exponential Euler, and semi-implicit Euler) commonly used in computational neuroscience, with no increase in computational cost. These limit cycles are important to preserve, due to their role in neuronal spiking. The second-order Strang splitting method is seen to perform especially well across a range of non-stiff and stiff dynamics.2010 Mathematics Subject Classification. 65P30, 37M20, 92C20. 1 It is straightforward to generalize what follows to the case where xi and bi are vector-valued and ai is matrix-valued, so that xi satisfies a first-order linear system of ODEs with constant coefficients when xj is stationary for j = i, but the scalar case covers the applications we are interested in.

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Section: Numerical Integrators For Conditionally Linear Systemsmentioning

confidence: 99%

Structure-Preserving Numerical Integrators for Hodgkin--Huxley-Type Systems

Chen¹,

Raman²,

Stern³

2020

SIAM J. Sci. Comput.

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“…For the Störmer-Verlet method, which can be interpreted as a trigonometric integrator with modified frequencies, related long-time results are given in [HL00a,CGHL15]. The numerical version of the energy transfer of property (iii) was studied in [HLW02, Section XIII.4] and in [CHL05,MS14]. Getting the energy transfer qualitatively correct by the numerical method turns out to put more restrictions on the choice of methods than long-time energy conservation.…”

Section: 22mentioning

confidence: 99%

Dynamics, Numerical Analysis, and Some Geometry

Gauckler¹,

Hairer²,

Lubich

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of time-dependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.

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“…In mechanical systems, energy conservation (or Hamiltonian system) is studied along with the symplectic numerical integrators which is the subtopic of geometric numerical integrators [MS14], [SD06]. Several papers such as [HL00], [SG09], [LMOW04] studied symplectic integration systems for long‐time energy conservation.…”

Section: Related Workmentioning

confidence: 99%

“…In this paper, we have utilized a simplified version of graph coloring algorithm which only addresses to group the non-adjacent primitives of the mesh similar to [FP14]. In mechanical systems, energy conservation (or Hamiltonian system) is studied along with the symplectic numerical integrators which is the subtopic of geometric numerical integrators [MS14], [SD06]. Several papers such as [HL00], [SG09], [LMOW04] studied symplectic integration systems for long-time energy conservation.…”

Section: Related Workmentioning

confidence: 99%

Position‐Based Simulation of Elastic Models on the GPU with Energy Aware Gauss‐Seidel Algorithm

Cetinaslan

2019

Computer Graphics Forum

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In this paper, we provide a smooth extension of the energy aware Gauss‐Seidel iteration to the Position‐Based Dynamics (PBD) method. This extension is inspired by the kinetic and potential energy changes equalization and uses the foundations of the recent extended version of PBD algorithm (XPBD). The proposed method is not meant to conserve the total energy of the system and modifies each position constraint based on the equality of the kinetic and potential energy changes within the Gauss‐Seidel process of the XPBD algorithm. Our extension provides an implicit solution for relatively better stiffness during the simulation of elastic objects. We apply our solution directly within each Gauss‐Seidel iteration and it is independent of both simulation step‐size and integration methods. To demonstrate the benefits of our proposed extension with higher frame rates, we develop an efficient and practical mesh coloring algorithm for the XPBD method which provides parallel processing on a GPU. During the initialization phase, all mesh primitives are grouped according to their connectivity. Afterwards, all these groups are computed simultaneously on a GPU during the simulation phase. We demonstrate the benefits of our method with many spring potential and strain‐based continuous material constraints. Our proposed algorithm is easy to implement and seamlessly fits into the existing position‐based frameworks.

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Modified Trigonometric Integrators

Cited by 27 publications

References 18 publications

Structure-Preserving Numerical Integrators for Hodgkin--Huxley-Type Systems

Structure-Preserving Numerical Integrators for Hodgkin--Huxley-Type Systems

Dynamics, Numerical Analysis, and Some Geometry

Position‐Based Simulation of Elastic Models on the GPU with Energy Aware Gauss‐Seidel Algorithm

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