Summation by parts, projections, and stability. I

Olsson, Pelle

doi:10.1090/s0025-5718-1995-1297474-x

Cited by 141 publications

(91 citation statements)

References 4 publications

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“…In this work, we develop a hybrid finite volume / finite difference method based on summation-by-parts (SBP) difference operators [11,34,35,50,55]. Other hybrid methods, combining finite elements and finite differences, have been developed for earthquake rupture dynamics (e.g., [6,37,40]), but require the use of overlapping grids and have no associated proofs of stability or accuracy.…”

Section: Introductionmentioning

confidence: 99%

Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

OReilly

Nordström

Kozdon

et al. 2015

Commun. Comput. Phys.

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Abstract.A numerical method suitable for wave propagation problems in complex geometries is developed for simulating dynamic earthquake ruptures with realistic friction laws. The numerical method couples an unstructured, node-centered finite volume method to a structured, high order finite difference method. In this work we our focus attention on 2-D antiplane shear problems. The finite volume method is used on unstructured triangular meshes to resolve earthquake ruptures propagating along a nonplanar fault. Outside the small region containing the geometrically complex fault, a high order finite difference method, having superior numerical accuracy, is used on a structured grid. The finite difference method is coupled weakly to the finite volume method along interfaces of collocated grid points. Both methods are on summation-by-parts form. The simultaneous approximation term method is used to weakly enforce the interface conditions. At fault interfaces, fault strength is expressed as a nonlinear function of sliding velocity (the jump in particle velocity across the fault) and a state variable capturing the history dependence of frictional resistance. Energy estimates are used to prove that both types of interface conditions are imposed in a stable manner. Stability and accuracy of the numerical implementation are verified through numerical experiments, and efficiency of the hybrid approach is confirmed through grid coarsening tests. Finally, the method is used to study earthquake rupture propagation along the margins of a volcanic plug. AMS subject classifications: 35L05,35L65,35Q35,65M06,65M08,65M12,65Z05

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Section: Introductionmentioning

confidence: 99%

Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

OReilly

Nordström

Kozdon

et al. 2015

Commun. Comput. Phys.

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“…The PC operators are modified at boundaries in a stable way by the summation-by-part (SBP) operators [17,16,32]. If D b is the standard pth order SBP for the centered difference operators, then the pth order PC operators are modified in this way …”

Section: Cen4mentioning

confidence: 99%

Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows

Wang¹,

Yee²,

Sjögreen³

et al. 2011

Journal of Computational Physics

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The goal of this paper is to generalize the well-balanced approach for non-equilibrium flow studied by Wang et al. [26] to a class of low dissipative high order shock-capturing filter schemes and to explore more advantages of well-balanced schemes in reacting flows. The class of filter schemes developed by Yee et al. [30], Sjögreen & Yee [24] and Yee & Sjögreen [35] consist of two steps, a full time step of spatially high order non-dissipative base scheme and an adaptive nonlinear filter containing shock-capturing dissipation. A good property of the filter scheme is that the base scheme and the filter are stand alone modules in designing. Therefore, the idea of designing a well-balanced filter scheme is straightforward, i.e., choosing a well-balanced base scheme with a well-balanced filter (both with high order). A typical class of these schemes shown in this paper is the high order central difference schemes/predictor-corrector (PC) schemes with a high order well-balanced WENO filter. The new filter scheme with the well-balanced property will gather the features of both filter methods and well-balanced properties: it can preserve certain steady state solutions exactly; it is able to capture small perturbations, e.g., turbulence fluctuations; it adaptively controls numerical dissipation. Thus it shows high accuracy, efficiency and stability in shock/turbulence interactions. Numerical examples containing 1D and 2D smooth problems, 1D stationary contact discontinuity problem and 1D turbulence/shock interactions are included to verify the improved accuracy, in addition to the well-balanced behavior.

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“…These SBP operators in combination with weak well-posed boundary conditions lead to energy stability [6,8,16,19,31,40,41]. One such boundary treatment is the simultaneous approximation term (SAT) method [7], which linearly combines the partial differential equation to be solved with well-posed boundary conditions [5,8,34,39].…”

Section: Introductionmentioning

confidence: 99%

A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

Nordström

Abbas

Erickson

et al. 2014

Commun. comput. phys.

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Abstract.A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.

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Summation by parts, projections, and stability. I

Cited by 141 publications

References 4 publications

Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

Simulation of Earthquake Rupture Dynamics in Complex Geometries Using Coupled Finite Difference and Finite Volume Methods

Construction of low dissipative high-order well-balanced filter schemes for non-equilibrium flows

A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

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