Finite volume methods, unstructured meshes and strict stability for hyperbolic problems

Nordström, Jan; Forsberg, Karl; Adamsson, Carl; Eliasson, Peter

doi:10.1016/s0168-9274(02)00239-8

Cited by 113 publications

(98 citation statements)

References 27 publications

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“…Figures 3a-b shows the construction of control volumes. A complete derivation of the finite volume operators are given in [44]. The finite volume approximation to the integral form of the momentum balance equation in (2.1) is written as…”

Section: Finite Volume Methodsmentioning

confidence: 99%

“…Thus, we take a different approach here and instead use an unstructured, node-centered finite volume method around the geometric complexity. The node-centered finite-volume method was shown to be an SBP scheme in [44] and has been coupled to high order finite difference methods for both advection [45] and advection-diffusion [21,48] problems. Here we extend the coupling to the scalar wave equation (written in first order form for velocities and stresses).…”

Section: Introductionmentioning

confidence: 99%

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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: Finite Volume Methodsmentioning

confidence: 99%

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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“…We stress that any discretization technique that can be formulated on SBP form such as for example finite difference [7,8], finite volume [9,10], spectral element [11,12], discontinuous Galerkin [13,14] and flux reconstruction schemes [15,16] will lead to the same analysis and principal results.…”

Section: Remark 51mentioning

confidence: 99%

“…It has been shown previously that a weak imposition of well-posed boundary conditions for finite difference [7,8], finite volume [9,10], spectral element [11,12], discontinuous Galerkin [13,14] and flux reconstruction schemes [15,16] on summation-by-parts (SBP) form can lead to energy stability. We will show that the continuous analysis of well posed boundary conditions implemented with weak boundary procedures together with schemes on Summation-by-parts (SBP) form automatically leads to stability.…”

Section: Introductionmentioning

confidence: 99%

A Roadmap to Well Posed and Stable Problems in Computational Physics

Nordström

2016

J Sci Comput

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All numerical calculations will fail to provide a reliable answer unless the continuous problem under consideration is well posed. Well-posedness depends in most cases only on the choice of boundary conditions. In this paper we will highlight this fact, and exemplify by discussing well-posedness of a prototype problem: the time-dependent compressible Navier-Stokes equations. We do not deal with discontinuous problems, smooth solutions with smooth and compatible data are considered. In particular, we will discuss how many boundary conditions are required, where to impose them and which form they should have in order to obtain a well posed problem. Once the boundary conditions are known, one issue remains; they can be imposed weakly or strongly. It is shown that the weak and strong boundary procedures produce similar continuous energy estimates. We conclude by relating the well-posedness results to energy-stability of a numerical approximation on summation-by-parts form. It is shown that the results obtained for weak boundary conditions in the well-posedness analysis lead directly to corresponding stability results for the discrete problem, if schemes on summation-by-parts form and weak boundary conditions are used. The analysis in this paper is general and can without difficulty be extended to any coupled system of partial differential equations posed as an initial boundary value problem coupled with a numerical method on summation-by parts form with weak boundary conditions. Our ambition in this paper is to give a general roadmap for how to construct a well posed continuous problem and a stable numerical approximation, not to give exact answers to specific problems.

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“…The main reason to use weak boundary procedures stems from the fact that together with summation-by-parts operators they lead to provable stable schemes. For application of this technique to finite difference methods, node-centered finite volume methods, spectral domain methods and various hybrid methods see [25,42,4,30,31,36,44,48,20,39,6,22,13,24], [32,47,45,46,14,41], [18,16,19,7] and [33,34,15,37,5] respectively. In this paper we will consider a new effect of using weak boundary procedures, namely that it in many cases (all that we tried) speeds up the convergence to steady-state.…”

Section: Introductionmentioning

confidence: 99%

Weak and strong wall boundary procedures and convergence to steady-state of the Navier–Stokes equations

Nordström

Eriksson

Eliasson

2012

Journal of Computational Physics

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Finite volume methods, unstructured meshes and strict stability for hyperbolic problems

Cited by 113 publications

References 27 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

A Roadmap to Well Posed and Stable Problems in Computational Physics

Weak and strong wall boundary procedures and convergence to steady-state of the Navier–Stokes equations

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