DeC and ADER: Similarities, Differences and a Unified Framework

Veiga, Maria Han; Öffner, Philipp; Torlo, Davide

doi:10.1007/s10915-020-01397-5

Cited by 25 publications

(19 citation statements)

References 36 publications

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“…Finally, extensions to relaxation approaches introduced by Ketcheson et al [12,21] are also possible to construct entropy conservative ADER schemes, where this is already work in progress for the DeC approach. Finally, in spirit of open science, all codes used to produce this paper are available in [9].…”

Section: Discussionmentioning

confidence: 99%

DeC and ADER: Similarities, Differences and a Unified Framework

Veiga¹,

Öffner²,

Torlo³

2020

Preprint

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In this paper, we demonstrate that the ADER approach as it is used inter alia in [29] can be seen as a special interpretation of the deferred correction (DeC) method as introduced in [6]. By using this fact, we are able to embed ADER in a theoretical background of time integration schemes and prove the relation between the accuracy order and the number of iterations which are needed to reach the desired order. Finally, we can also investigate the stability regions for the ADER approach for different orders using several basis functions and compare them with the DeC ansatz.

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Section: Discussionmentioning

confidence: 99%

DeC and ADER: Similarities, Differences and a Unified Framework

Veiga¹,

Öffner²,

Torlo³

2020

Preprint

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“…However, this part should clarify the connection between DeC and classical RK schemes. Due to the close connection between the modern ADER approach and DeC [19], this will also open the theoretical investigations of ADER. Next, we focus on the relaxation approach.…”

Section: The Connection Between Dec and Rk With Possible Application ...mentioning

confidence: 94%

“…After describing the general construction of RD and DeC schemes, they can be coupled to a explicit spacetime FE scheme for the initial value problem (1). We like to point out that this approach has several similarities and connections to the modern ADER approach [19].…”

Section: Residual Distribution For Hyperbolic Problemsmentioning

confidence: 99%

Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

Abgrall¹,

Mélédo²,

Öffner³

et al. 2021

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In [2] is proposed a simplified DeC method, that, when combined with the residual distribution (RD) framework, allows to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix for hyperbolic problems. In this paper, we close some open gaps in the context of deferred correction (DeC) and their application within the RD framework. First, we demonstrate the connection between the DeC schemes and the RK methods. With this knowledge, DeC can be rewritten as a convex combination of explicit Euler steps, showing the connection to the strong stability preserving (SSP) framework. Then, we can apply the relaxation approach introduced in [22] and construct entropy conservative/dissipative DeC (RDeC) methods, using the entropy correction function proposed in [3].

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“…Hence, the DeC algorithm can be seen as Algorithm 1 where (38) gives the update formula. For more information and properties of the DeC approach, we refer to [3,31] and references therein. In the following, we explain how to adapt the DeC approach to obtain a conservative and positivity preserving time integration scheme.…”

Section: Deferred Correction Methodsmentioning

confidence: 99%

“…Many problems (31) in nature can be written as a production destruction system (PDS) for the unknown…”

Section: Patankar Methods For Production-destruction Systemsmentioning

confidence: 99%

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

Ciallella¹,

Lorenzo²,

Öffner³

et al. 2021

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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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DeC and ADER: Similarities, Differences and a Unified Framework

Cited by 25 publications

References 36 publications

DeC and ADER: Similarities, Differences and a Unified Framework

DeC and ADER: Similarities, Differences and a Unified Framework

Relaxation Deferred Correction Methods and their Applications to Residual Distribution Schemes

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

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