Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes

Öffner, Philipp; Torlo, Davide

doi:10.1016/j.apnum.2020.01.025

Cited by 40 publications

(67 citation statements)

References 14 publications

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“…In fact, as extensively reported in the literature, the use of positive integrators much improve the results whenever positive quantities (i.e. density, concentration) need to be numerically simulated [52] , [53] , [54] , [55] , [56] , [57] .…”

Section: Appendix Bmentioning

confidence: 70%

Using awareness to Z-control a SEIR model with overexposure: Insights on Covid-19 pandemic

Lacitignola

Diele

2021

Chaos, Solitons & Fractals

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In this paper, we use the Z-control approach to get further insight on the role of awareness in the management of epidemics that, just like Covid-19, display a high rate of overexposure because of the large number of asymptomatic people. We focus on a SEIR model including a overexposure mechanism and consider awareness as a time-dependent variable whose dynamics is not assigned a priori. Exploting the potential of awareness to produce social distancing and self-isolation among susceptibles, we use it as an indirect control on the class of infective individuals and apply the Z-control approach to detect what trend must awareness display over time in order to eradicate the disease. To this aim, we generalize the Z-control procedure to appropriately treat an uncontrolled model with more than two governing equations. Analytical and numerical investigations on the resulting Z-controlled system show its capability in controlling some representative dynamics within both the backward and the forward scenarios. The awareness variable is qualitatively compared to Google Trends data on Covid-19 that are discussed in the perspective of the Z-control approach, inferring qualitative indications in view of the disease control. The cases of Italy and New Zealand in the first phase of the pandemic are analyzed in detail. The theoretical framework of the Z-control approach can hence offer the chance to reflect on the use of Google Trends as a possible indicator of good management of the epidemic.

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Section: Appendix Bmentioning

confidence: 70%

Using awareness to Z-control a SEIR model with overexposure: Insights on Covid-19 pandemic

Lacitignola

Diele

2021

Chaos, Solitons & Fractals

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“…The time discretization is one of the major point of this paper since we want to apply for the first time the arbitrary high-order, conservative and positivity preserving modified Patankar Deferred Correction method (mPDeC) together with the described WENO approach to the shallow water equations resulting in a high-order, conservative, positivity preserving, non-oscillatory and well-balanced scheme. Before describing the combined algorithm, we introduce the Deferred Correction (DeC) method [22] as described in [1,6,61] and we repeat its modification using the Patankar trick from [48,47,62].…”

Section: Time Discretizationmentioning

confidence: 99%

“…Even if (modified) Patankar (mP) methods have been already used inside a numerical method for fluid simulations [32,33,42], those mP schemes have been based on extensions of classical RK methods and they are of maximum order three 1 . By applying the modified Patankar trick inside the Deferred Correction (DeC) framework, the authors of [48] were able to construct a conservative, arbitrarily high-order and positivity preserving method for production-destruction systems (PDS) of ordinary differential equations. To obtain a positive WENO spatial reconstruction, a positive limiter must be used [66,50].…”

Section: Introductionmentioning

confidence: 99%

“…However, all the ingredients to go to arbitrary high order can be found in the related repository [19]. In Section 4, the time-integration is considered focusing on Deferred Correction (DeC), the modified Patankar approach and its combination to the modified Patankar DeC (mPDeC) method developed in [48]. In Section 5, we describe how mPDeC can be combined with the WENO approach.…”

Section: Introductionmentioning

confidence: 99%

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An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

Ciallella¹,

Lorenzo²,

Öffner³

et al. 2021

Preprint

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In this paper, we develop and present an arbitrary high order well-balanced finite volume WENO method combined with the modified Patankar Deferred Correction (mPDeC) time integration method for the shallow water equations. Due to the positivity-preserving property of mPDeC, the resulting scheme is unconditionally positivity preserving for the water height. To apply the mPDeC approach, we have to interpret the spatial semi-discretization in terms of production-destruction systems. Only small modifications inside the classical WENO implementation are necessary and we explain how it can be done. In numerical simulations, focusing on a fifth order method, we demonstrate the good performance of the new method and verify the theoretical properties.

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“…The DeC scheme is an algorithm that allows us to obtain a high order scheme starting from a low order one in a general way. It has already been used for implicit schemes in ODE and PDE contexts, as well as in combination with RK schemes; see, for example, [11,22].…”

Section: B826mentioning

confidence: 99%

High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models

Abgrall¹,

Torlo²

2020

SIAM J. Sci. Comput.

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This work introduces an extension of the residual distribution (RD) framework to stiff relaxation problems. The RD is a class of schemes which is used to solve a hyperbolic system of partial differential equations. To our knowledge, it has been used only for systems with mild source terms, such as gravitation problems or shallow water equations. What we propose is an implicit-explicit (IMEX) version of the RD schemes that can resolve stiff source terms, without refining the discretization up to the stiffness scale. This can be particularly useful in various models, where the stiffness is given by topological or physical quantities, e.g., multiphase flows, kinetic models, or viscoelasticity problems. We will focus on kinetic models that are BGK approximation of hyperbolic conservation laws. The extension to more complicated problems will be carried out in future works. The provided scheme is able to catch different relaxation scales automatically, without losing accuracy; we prove that the scheme is asymptotic preserving and this guarantees that, in the relaxation limit, we recast the expected macroscopic behavior. To get a high order accuracy, we use an IMEX time discretization combined with a deferred correction procedure, while naturally RD provides high order space discretization. Finally, we show some numerical tests in one and two dimensions for stiff systems of equations.

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Arbitrary high-order, conservative and positivity preserving Patankar-type deferred correction schemes

Cited by 40 publications

References 14 publications

Using awareness to Z-control a SEIR model with overexposure: Insights on Covid-19 pandemic

Using awareness to Z-control a SEIR model with overexposure: Insights on Covid-19 pandemic

An Arbitrary High Order and Positivity Preserving Method for the Shallow Water Equations

High Order Asymptotic Preserving Deferred Correction Implicit-Explicit Schemes for Kinetic Models

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