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Calvo, M.; Montijano, J. I.; Rández, L.

doi:10.1023/a:1025507920426

Cited by 9 publications

(4 citation statements)

References 11 publications

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“…Time stepping methods usually rely on local error estimators to detect and locate discontinuities: a large value of the local error estimate indicates the presence of a discontinuity and different strategies to estimate local errors are available (Shampine & Watts 1971;Butcher & Johnston 1993). Alternatively, Calvo et al (2003) exposed the possibility to detect discontinuities by using the so-called pairs of embedded forms or by monitoring the size of an estimate of the local defect. Some discontinuities might avoid detection.…”

Section: Time Stepping Methodsmentioning

confidence: 99%

“…Discontinuous differential systems frequently appear in various fields, such as chemical kinetics (e.g., Landry et al 2009), biological systems (e.g., Gouzé & Sari 2002), and various engineering disciplines (e.g., Malmborg & Bernhardsson 1997;Acary & Brogliato 2008). Moreover, most of the studies on discontinuous ODEs rely on numerical methods and a broad literature attests to the importance of this topic: from the pioneering works by Chartres & Stepleman (1972) and Mannshardt (1978) to the surveys by Calvo et al (2008) and Dieci & Lopez (2012).…”

Section: Ordinary Differential Equations With a Discontinuous Right-h...mentioning

confidence: 99%

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Discontinuities in numerical radiative transfer

Janett

2019

A&A

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Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of the stationary radiative transfer equation is particularly challenging in such situations, because standard numerical methods may perform very inefficiently in the absence of local smoothness. However, a rigorous investigation of the numerical treatment of the radiative transfer equation in discontinuous media is still lacking. The aim of this work is to expose the limitations of standard convergence analyses for this problem and to identify the relevant issues. Moreover, specific numerical tests are performed. These show that discontinuities in the atmospheric physical parameters effectively induce first-order discontinuities in the radiative transfer equation, reducing the accuracy of the solution and thwarting high-order convergence. In addition, a survey of the existing numerical schemes for discontinuous ordinary differential systems and interpolation techniques for discontinuous discrete data is given, evaluating their applicability to the radiative transfer problem.

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Section: Time Stepping Methodsmentioning

confidence: 99%

Section: Ordinary Differential Equations With a Discontinuous Right-h...mentioning

confidence: 99%

Discontinuities in numerical radiative transfer

Janett

2019

A&A

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“…At a jump, adaptive ODE solvers will continuously reduce step size in response to increased error, possibly until numerical underflow occurs. Even if integration is possible across the jump, it is slow, and the global error of the solution can be adversely affected (Calvo et al, 2003). Typically, these issues can be easily avoided by restarting ODE solutions at the discontinuity but this requires these positions to be known.…”

Section: Representational Limitations Of the Neural Odementioning

confidence: 99%

Segmenting Hybrid Trajectories using Latent ODEs

Shi¹,

Morris²

2021

Preprint

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Smooth dynamics interrupted by discontinuities are known as hybrid systems and arise commonly in nature. Latent ODEs allow for powerful representation of irregularly sampled time series but are not designed to capture trajectories arising from hybrid systems. Here, we propose the Latent Segmented ODE (LatSegODE), which uses Latent ODEs to perform reconstruction and changepoint detection within hybrid trajectories featuring jump discontinuities and switching dynamical modes. Where it is possible to train a Latent ODE on the smooth dynamical flows between discontinuities, we apply the pruned exact linear time (PELT) algorithm to detect changepoints where latent dynamics restart, thereby maximizing the joint probability of a piece-wise continuous latent dynamical representation. We propose usage of the marginal likelihood as a score function for PELT, circumventing the need for modelcomplexity-based penalization. The LatSegODE outperforms baselines in reconstructive and segmentation tasks including synthetic data sets of sine waves, Lotka Volterra dynamics, and UCI Character Trajectories.

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“…Recently, standard techniques to study discontinuous systems of ODEs have been employed in the simulation of reaction diffusion systems with discontinuous forcing terms (see for instance [4,2,3]). The solutions of such systems can be simulated using suitable numerical methods, see for instance the book [1] or the papers [10,11,37].…”

mentioning

confidence: 99%

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

Colombo¹,

Buono²,

Lopez³

et al. 2018

Discrete &Amp; Continuous Dynamical Systems - B

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Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

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Untitled

Cited by 9 publications

References 11 publications

Discontinuities in numerical radiative transfer

Discontinuities in numerical radiative transfer

Segmenting Hybrid Trajectories using Latent ODEs

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems

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