Summation-by-parts in time

Nordström, Jan; Lundquist, Tomas

doi:10.1016/j.jcp.2013.05.042

Cited by 82 publications

(134 citation statements)

References 42 publications

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“…In time we use L time levels from 0 to T. The first derivative u ξ is approximated by D ξ u, where D ξ is a so-called SBP operator, see [10]. A multi-dimensional finite difference approximation (including the time discretization [8,4]), on SBP-SAT form, is constructed by extending the one-dimensional SBP operators in a tensor product fashion as…”

Section: The Discrete Problemmentioning

confidence: 99%

“…The energy method (multiplying from the left with V T (P τ ⊗ P ξ ⊗ P η ⊗ I)) is applied to (8) and the equation is added to its transpose. The result is…”

Section: Stabilitymentioning

confidence: 99%

“…In this paper (and the full paper [5]) we treat the time-dependent transformations in a SBP-SAT framework. To guarantee stability of the fully discrete approximation we employ the recently developed SBP-SAT technique in time [8,4].…”

Section: Introductionmentioning

confidence: 99%

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Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains

Nikkar

Nordström

2015

Lecture Notes in Computational Science and Engineering

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A time-dependent coordinate transformation of a constant coefficient hyperbolic system of equations is considered. We use the energy method to derive well-posed boundary conditions for the continuous problem. Summation-by-Parts (SBP) operators together with a weak imposition of the boundary and initial conditions using Simultaneously Approximation Terms (SATs) guarantee energy-stability of the fully discrete scheme. We construct a time-dependent SAT formulation that automatically imposes the boundary conditions, and show that the numerical Geometric Conservation Law (GCL) holds. Numerical calculations corroborate the stability and accuracy of the approximations. As an application we study the sound propagation in a deforming domain using the linearized Euler equations.

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Section: The Discrete Problemmentioning

confidence: 99%

“…The energy method (multiplying from the left with V T (P τ ⊗ P ξ ⊗ P η ⊗ I)) is applied to (8) and the equation is added to its transpose. The result is…”

Section: Stabilitymentioning

confidence: 99%

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Fully Discrete Energy Stable High Order Finite Difference Methods for Hyperbolic Problems in Deforming Domains

Nikkar

Nordström

2015

Lecture Notes in Computational Science and Engineering

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“…Such schemes typically generate a linear system of ordinary differential equations, which is integrated in time using explicit methods. The groundwork for employing SBP-SAT also as a method of time integration was laid in [10]. However, naive usage of SBP in time produces schemes that, while provably stable and high order accurate, lead to large systems which are difficult to solve efficiently in multiple dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…We consider provably stable SBP based domain decomposition methods for a two-dimensional advection-diffusion problem, where local solutions are coupled at the subdomain interfaces using SATs. The coupling procedure follows the ideas in [1], with adjustments to account for the use of SBP in time [7,10]. Our scheme involves isolating a linear system consisting only of interface components by solving independent, subdomain sized systems.…”

Section: Introductionmentioning

confidence: 99%

A Stable Domain Decomposition Technique for Advection–Diffusion Problems

Ålund

Nordström

2018

J Sci Comput

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The use of implicit methods for numerical time integration typically generates very large systems of equations, often too large to fit in memory. To address this it is necessary to investigate ways to reduce the sizes of the involved linear systems. We describe a domain decomposition approach for the advection-diffusion equation, based on the Summationby-Parts technique in both time and space. The domain is partitioned into non-overlapping subdomains. A linear system consisting only of interface components is isolated by solving independent subdomain-sized problems. The full solution is then computed in terms of the interface components. The Summation-by-Parts technique provides a solid theoretical framework in which we can mimic the continuous energy method, allowing us to prove both stability and invertibility of the scheme. In a numerical study we show that single-domain implementations of Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to compute solutions for grid resolutions that cannot be handled efficiently using a single-domain formulation. An order of magnitude speed-up is observed, both compared to a single-domain formulation and to explicit Runge-Kutta time integration.

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