On pseudo-spectral time discretizations in summation-by-parts form

Total variation diminishing multigrid methods have been developed for first order accurate discretizations of hyperbolic conservation laws. This technique is based on a so-called upwind biased residual interpolation and allows for algorithms devoid of spurious numerical oscillations in the transient phase. In this paper, we justify the introduction of such prolongation and restriction operators by rewriting the algorithm in a matrix-vector notation. This perspective sheds new light on multigrid procedures for hyperbolic problems and provides a direct extension for high order accurate difference approximations. The new multigrid procedure is presented, advantages and disadvantages are discussed and numerical experiments are performed.

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“…If the eigenvalues of L 1 have strictly positive real parts, then the convergence of (9) is guaranteed [9,10].…”

Section: Convergence To Steady-state For Single Grid Methodsmentioning

confidence: 99%

“…The invertibility of L 1 can be shown for pseudo-spectral approximations [9], but not in general for discretizations based on finite-difference methods [11]. The 1st and 2nd order upwind SBP operators lead to matrices L 1 with triangular and block-triangular structure, respectively, for which invertibility follows in a straightforward way.…”

Section: Remarkmentioning

confidence: 99%

Multigrid Schemes for High Order Discretizations of Hyperbolic Problems

Ruggiu

Nordström

2020

J Sci Comput

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“…Turning to SBP methods in time [2,16,21], a class of linearly and nonlinearly stable SBP schemes has been constructed and studied in this context, see also [15,36,37]. If the underlying quadrature is chosen as Radau or Lobatto quadrature, these Runge-Kutta schemes are exactly the classical Radau IA, Radau IIA, and Lobatto IIIC methods [26].…”

Section: Introductionmentioning

confidence: 99%

Some notes on summation by parts time integration methods

Ranocha

2019

Results in Applied Mathematics

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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 necessarily involved in provably stable numerical methods. This article contributes to this topic by proposing a novel class of A stable SBP time integration methods including 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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“…Summation-by-parts (SBP) operators [12,21], with simultaneousapproximation-terms (SAT) weakly imposing boundary and initial conditions, allow for energy-stable and high order accurate approximations [6,17,14,3]. The solution of the resulting fully discrete problem is unique, provided that the eigenvalues of the time discretization operator have strictly positive real parts [17,14,18]. This assumption on the eigenvalues has been proved for pseudospectral collocation methods [19] and second order finite difference discretizations [17].…”

mentioning

confidence: 99%

“…This assumption on the eigenvalues has been proved for pseudospectral collocation methods [19] and second order finite difference discretizations [17]. However, it does not hold for all types of SBP-SAT approximations [18,13], and it has only been conjectured for higher order finite difference methods [17,14].…”

mentioning

confidence: 99%

Eigenvalue Analysis for Summation-by-Parts Finite Difference Time Discretizations

Ruggiu¹,

Nordström²

2020

SIAM J. Numer. Anal.

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Diagonal norm finite difference based time integration methods in summation-byparts form are investigated. The second, fourth, and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully discrete approximations of initial boundary value problems. Our findings also allow us to conclude that the Runge-Kutta methods based on second, fourth, and sixth order summation-by-parts finite difference time discretizations automatically satisfy previously unreported stability properties. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil.

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On pseudo-spectral time discretizations in summation-by-parts form

Cited by 12 publications

References 25 publications

Multigrid Schemes for High Order Discretizations of Hyperbolic Problems

Multigrid Schemes for High Order Discretizations of Hyperbolic Problems

Some notes on summation by parts time integration methods

Eigenvalue Analysis for Summation-by-Parts Finite Difference Time Discretizations

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