An adaptive local mesh refinement method for time-dependent partial differential equations

Arney, David C.; Flaherty, Joseph E.

doi:10.1016/0168-9274(89)90011-1

Cited by 32 publications

(28 citation statements)

References 13 publications

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“…These are the same as in [11]. For related earlier work on LUGR methods, we refer to Berger and Öliger [3], Gropp [6,7], Arney and Flaherty [2], and references therein.…”

Section: Introductionmentioning

confidence: 96%

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Runge-Kutta methods and local uniform grid refinement

Trompert¹,

Verwer²

1993

Math. Comp.

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Abstract. Local uniform grid refinement (LUGR) is an adaptive grid technique for computing solutions of partial differential equations possessing sharp spatial transitions. Using nested, finer-and-finer uniform subgrids, the LUGR technique refines the space grid locally around these transitions, so as to avoid discretization on a very fine grid covering the entire physical domain. This paper examines the LUGR technique for time-dependent problems when combined with static regridding. Static regridding means that in the course of the time evolution, the space grid is adapted at discrete times. The present paper considers the general class of Runge-Kutta methods for the numerical time integration. Following the method of lines approach, we develop a mathematical framework for the general Runge-Kutta LUGR method applied to multispace-dimensional problems. We hereby focus on parabolic problems, but a considerable part of the examination applies to hyperbolic problems as well. Much attention is paid to the local error analysis. The central issue here is a "refinement condition" which is to underly the refinement strategy. By obeying this condition, spatial interpolation errors are controlled in a manner that the spatial accuracy obtained is comparable to the spatial accuracy on the finest grid if this grid would be used without any adaptation. A diagonally implicit Runge-Kutta method is discussed for illustration purposes, both theoretically and numerically.

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“…These are the same as in [11]. For related earlier work on LUGR methods, we refer to Berger and Öliger [3], Gropp [6,7], Arney and Flaherty [2], and references therein.…”

Section: Introductionmentioning

confidence: 96%

“…by using (6.5), the nontrivial components of the spatial error function wk (Zk)~xDk(Tk are approximated by (2) (6.16)…”

Section: J3>mentioning

confidence: 99%

Runge-Kutta methods and local uniform grid refinement

Trompert¹,

Verwer²

1993

Math. Comp.

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“…Such composite grids can be found in e.g. [1,3,5,[10][11][12]18]. The grids we consider result from a uniform basis grid with meshsize H, cf.…”

Section: Composite Gridsmentioning

confidence: 99%

A finite volume scheme for solving elliptic boundary value problems on composite grids

1998

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. (1998). A finite volume scheme for solving elliptic boundary value problems on composite grids. (RANA : reports on applied and numerical analysis; Vol. 9804). Eindhoven: Technische Universiteit Eindhoven. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

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“…The utility of such adaptive techniques is greatly enhanced when they are capable of providing an estimate of the accuracy of the computed solution. Local error estimates are often used as refinement indicators and to produce solutions that satisfy either local or global accuracy specifications [1,2,3,6,14,15,26). Successful error estimates have been obtained using h-refinement [6,14,15], where the difference between solutions on different meshes is used to estimate the error, and p-refinement [1,2,3,15,211 where the difference between methods of different orders are used to estimate the error.…”

mentioning

confidence: 99%

“…Such techniques have been the subject of a great deal of recent attention (cf. Babuska et al [8,9) and Thompson [29]) and are generally capable of introducing finer-meshes in regions where greater resolution is needed [1,2,3,6,14,15,22,26], moving meshes in order to follow isolated dynamic phenomena [1,2,5,20,22,26,281, or changing the order of methods in specific regions of the problem domain [17,21]. The utility of such adaptive techniques is greatly enhanced when they are capable of providing an estimate of the accuracy of the computed solution.…”

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confidence: 99%

An adaptive mesh-moving and local refinement method for time-dependent partial differential equations

Arney

Flaherty

1990

ACM Trans. Math. Softw.

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We discuss mesh-moving, static mesh-regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time. A coarse base mesh of quadrilateral cells is moved by an algebraic mesh-movement function so as to follow and isolate spatially distinct phenomena. The local mesh-refinement method recursively divides the time step and spatial cells of the moving base mesh in regions where error indicators are high until a prescribed tolerance is satisfied. The static mesh-regeneration procedure is used to create a new base mesh when the existing one becomes too distorted. The adaptive methods have been combined with a MacCormack finite difference scheme for hyperbolic systems and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples.

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An adaptive local mesh refinement method for time-dependent partial differential equations

Cited by 32 publications

References 13 publications

Runge-Kutta methods and local uniform grid refinement

Runge-Kutta methods and local uniform grid refinement

A finite volume scheme for solving elliptic boundary value problems on composite grids

An adaptive mesh-moving and local refinement method for time-dependent partial differential equations

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