A simple adaptive technique for nonlinear wave problems

Sanz‐Serna, J. M.; Christie, I.

doi:10.1016/0021-9991(86)90267-6

Cited by 63 publications

(40 citation statements)

References 19 publications

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“…In practice, the adaptive solution process was implemented using MATLAB [16] and the COM-SOL Multiphysics finite element package [17]. The grid generation follows the method presented by Sanz-Serna and Christie in [18] which is equivalent to equidistributing the arc-length monitor function given by (10) 4 Analysis of mesh convergence…”

Section: Adaptive Solution Proceduresmentioning

confidence: 99%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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The Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. AbstractThis paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples.

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Section: Adaptive Solution Proceduresmentioning

confidence: 99%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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“…Concerning the grid determination, our algorithm can be classified as belonging to the class of methods which are "intermediate" between the static regridding methods, where nodes remain fixed for intervals of time [ 14,[22][23][24], and continuously moving grid methods, where the node movement and the PDE integration are fully coupled [2,10,17,18,20,26]. We have successfully applied this "intermediate" approach in [5,6].…”

Section: The "Intermediate" Approachmentioning

confidence: 99%

“…These schemes are "intermediate" between the static regridding methods [ 14,[22][23][24], where nodes remain fixed for intervals of time, and continuously moving grid methods, where the node movement and the PDE integration are fully coupled [2,10,17,18,20,26]. While the research in [5,6] has enabled us to identify a promising scheme, the implementation considered in those papers used fixed time-steps and did not allow a dynamic variation of the number of spatial grid-points.…”

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confidence: 99%

An adaptive moving grid method for one-dimensional systems of partial differential equations

Verwer¹,

Blom²,

Sanz‐Serna

1989

Journal of Computational Physics

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We describe a fully adaptive, moving grid method for solving initial-boundary value problems for systems of one-space dimensional partial differential equations whose solutions exhibit rapid variations in space and time. The method, based on finite-differences, is of the Lagrangian type and has been derived through a co-ordinate transformation which leads to equidistribution in space of the second derivative. Our technique is "intermediate" between static regridding methods, where nodes remain fixed for intervals of time, and continuously moving grid methods, where the node movement and the PDE integration are fully coupled. In our approach, the computation of the moving grids and the solution on these grids are carried out separately, while the nodes are moved at each time-step. Two error monitors have been implemented, one to govern the time-step selection and the other to eventually adapt the number of moving nodes. The method allows the use of different moving grids for different components in the PDE system. Numerical experiments are presented for a set of five sample problems from the literature, including two problems from combu~tion.,,.,

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“…In situations where the unknown u is a vector, the foregoing ideas still apply, provided that the bars in ( 3.3 )-(3.4) are understood to denote some suitable vector norm. It should be stressed that, although the introduction of the new variables (s, T) is helpful in the previous discussions, the computation of the grid {x;' }, at least for n "::3 1, can be completely achieved in terms of the old variables (Furzeland [7], Sanz-Serna and Christie [17]). To this effect it is enough to replace (3.2) by the midpoint quadrature, leading to the set of relations ( X;r1+X; )…”

Section: Equidistribution Of the Second Derivativementioning

confidence: 99%

“…Here the solution is advanced in time on a fixed nonuniform grid, while after each step, or series of steps, a regridding is carried out which is in turn followed by an interpolation to generate the initial values for the next step. Some recent contributions in this area are Bieterman and Babuska [2], Furzeland [7], Sanz-Serna and Christie [ 17 ], and Revilla [ 15].…”

Section: Introductionmentioning

confidence: 99%

On simple moving grid methods for one-dimensional evolutionary partial differential equations

Blom¹,

Sanz‐Serna

Verwer³

1988

Journal of Computational Physics

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Two moving grid algorithms for the numerical integration of evolutionary one-dimensional partial differential equations are considered. Both algorithms discretize the equation on trapezoidal space-time elements and combine a prediction step with a remeshing technique in order to determine the orientation of the sides of the trapezoids joining nodes at consecutive time levels. The behavior of the discretizations employed is forecast by means of the modified equation technique and then studied in a series of numerical experiments.

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A simple adaptive technique for nonlinear wave problems

Cited by 63 publications

References 19 publications

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

An adaptive moving grid method for one-dimensional systems of partial differential equations

On simple moving grid methods for one-dimensional evolutionary partial differential equations

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