An improved projection method

Mattsson, Ken; Olsson, Pelle

doi:10.1016/j.jcp.2018.06.030

Cited by 13 publications

(8 citation statements)

References 46 publications

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“…since t t t T L v v v is a scalar. Hence, (18) implies that v v v − t t t T L v v v1 1 1 ∈ V 0 . Moreover, (17) shows that an arbitrary u u u ∈ V 1 can be written as the image of a vector in V 0 under D. Therefore, D : V 0 → V 1 is surjective (i.e.…”

Section: By Definition the Adjoint Operatormentioning

confidence: 96%

“…Consequently, all A stable classical Runge-Kutta methods based on Radau and Lobatto quadrature rules can be formulated in the framework of SBP operators. Notably, instead of using simultaneous approximation terms (SATs) [4,5] to impose initial conditions weakly, these new schemes use a strong imposition of initial conditions in combination with a projection method [18,22,23].…”

Section: Introductionmentioning

confidence: 99%

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Some notes on summation by parts time integration methods

Ranocha

2019

Results in Applied Mathematics

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Section: By Definition the Adjoint Operatormentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Some notes on summation by parts time integration methods

Ranocha

2019

Results in Applied Mathematics

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“…Consequently, all A stable classical Runge-Kutta methods based on Radau and Lobatto quadrature rules can be formulated in the framework of SBP operators. Notably, instead of using simultaneous approximation terms (SATs) [5,6] to impose initial conditions weakly, these new schemes use a strong imposition of initial conditions in combination with a projection method [22,26,27].…”

Section: Introductionmentioning

confidence: 99%

A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Ranocha

Nordström

2021

J Sci Comput

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Since integration by parts is an important tool when deriving energy or entropy estimates for differential equations, one may conjecture that some form of summation by parts (SBP) property is involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods which can also be reformulated as implicit Runge-Kutta methods. In contrast to existing SBP time integration methods using simultaneous approximation terms to impose the initial condition weakly, the new schemes use a projection method to impose the initial condition strongly without destroying the SBP property. The new class of methods includes the classical Lobatto IIIA collocation method, not previously formulated as an SBP scheme. Additionally, a related SBP scheme including the classical Lobatto IIIB collocation method is developed.

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“…is approach is employed for approximating the derivatives in a spatial domain, together with the weak enforcement of boundaries and interface conditions, using either the simultaneous-approximationterm (SAT) method [20,21] or the projection method [22][23][24]. e SBP-SAT method can develop strictly stable proof using the energy method, leading to robustness in spatial and time domains.…”

Section: Introductionmentioning

confidence: 99%

Multiblock SBP-SAT Methodology of Symmetric Matrix Form of Elastic Wave Equations on Curvilinear Grids

Sun

Yang

Jiang

et al. 2020

Shock and Vibration

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A stable and accurate finite-difference discretization of first-order elastic wave equations is derived in this work. To simplify the origin and proof of the formulas, a symmetric matrix form (SMF) for elastic wave equations is presented. The curve domain is discretized using summation-by-parts (SBP) operators, and the boundary conditions are weakly enforced using the simultaneous-approximation-term (SAT) technique, which gave rise to a provably stable high-order SBP-SAT method via the energy method. In addition, SMF can be extended to wave equations of different types (SH wave and P-SV wave) and dimensions, which can simplify the boundary derivation process and improve its applicability. Application of this approximation can divide the domain into a multiblock context for calculation, and the interface boundary conditions of blocks can also be used to simulate cracks and other structures. Several numerical simulation examples, including actual elevation within the area of Lushan, China, are presented, which verifies the viability of the framework present in this paper. The applicability of simulating elastic wave propagation and the application potential in the seismic numerical simulation of this method are also revealed.

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An improved projection method

Cited by 13 publications

References 46 publications

Some notes on summation by parts time integration methods

Some notes on summation by parts time integration methods

A New Class of A Stable Summation by Parts Time Integration Schemes with Strong Initial Conditions

Multiblock SBP-SAT Methodology of Symmetric Matrix Form of Elastic Wave Equations on Curvilinear Grids

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