The numerical solution of Stefan problems with front-tracking and smoothing methods

Meyer, Gunter H.

doi:10.1016/0096-3003(78)90001-2

Cited by 14 publications

(8 citation statements)

References 20 publications

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“…The backward sweep Equation (2.11) also will be integrated with the trapezoidal rule. This method leads, of course, to a very crude resolution of each one dimensional two point boundary value problem, but numerous computations with fixed and free nonlinear boundary conditions have substantiated that for the slowly varying solutions typical of diffusion type problems the trapezoidal rule is accurate enough when answers from various mesh sizes are compared with analytical solutions of model problems (see [7] and [8]). The overriding advantage of the trapezoidal rule is the fact that no interpolation between mesh points is necessary during the reverse sweep which simplifies programming of the algorithm.…”

Section: Numerical Experimentsmentioning

confidence: 97%

“…It was shown in [7] and [8] that the method of lines-SOR-invariant imbedding method could reproduce numerically the analytic solutions of certain model problems with fixed and free boundaries. We shall discuss here its performance for two problems of comparable complexity without known solution in order to demonstrate the flexibility of the method and to verify some of the effects predicted by the theoretical results of the next section.…”

Section: Numerical Experimentsmentioning

confidence: 98%

“…The method of lines approximation for a general two dimensional elliptic operator subject to nonlinear or free boundary conditions on part of the boundary has been given in [7] and [8]. Here we shall merely summarize the algorithm for a special case, namely Poisson's equation on an annulus like domain ~ with Dirichlet data on the inner boundary 0~ 1 and general nonlinear data on the (possibly free) outer boundary ~%2.…”

Section: The Algorithmmentioning

confidence: 99%

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The method of lines for Poisson's equation with nonlinear or free boundary conditions

Meyer

1978

Numer. Math.

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Summary. The method of lines is used to solve Poisson's equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.

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Section: Numerical Experimentsmentioning

confidence: 97%

Section: Numerical Experimentsmentioning

confidence: 98%

Section: The Algorithmmentioning

confidence: 99%

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The method of lines for Poisson's equation with nonlinear or free boundary conditions

Meyer

1978

Numer. Math.

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“…The formalism can be adapted to accommodate time-varying bounds. This is more generally studied as a free boundary problem, with a typical example including the Stefan problem (Meyer, 1978;Li, 1997).…”

Section: Fokker-planck Formalism With Time-varying Boundsmentioning

confidence: 99%

A flexible framework for simulating and fitting generalized drift-diffusion models

Shinn

Lam

Murray

2020

eLife

102

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The drift-diffusion model (DDM) is an important decision-making model in cognitive neuroscience. However, innovations in model form have been limited by methodological challenges. Here, we introduce the generalized drift-diffusion model (GDDM) framework for building and fitting DDM extensions, and provide a software package which implements the framework. The GDDM framework augments traditional DDM parameters through arbitrary user-defined functions. Models are solved numerically by directly solving the Fokker-Planck equation using efficient numerical methods, yielding a 100-fold or greater speedup over standard methodology. This speed allows GDDMs to be fit to data using maximum likelihood on the full response time (RT) distribution. We demonstrate fitting of GDDMs within our framework to both animal and human datasets from perceptual decision-making tasks, with better accuracy and fewer parameters than several DDMs implemented using the latest methodology, to test hypothesized decision-making mechanisms. Overall, our framework will allow for decision-making model innovation and novel experimental designs.

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“…Due to the inherent non-linearity of the energy balance at the interface governing the interface position, few analytical solutions of interest are available [1][2][3][4] giving place to the development of a great number of numerical algorithms [5][6][7] based on finite differences [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] finite elements and, more recently, boundary elements [64][65][66][67]. All these methods can be classified into two main groups, front tracking methods and fixed domain methods.…”

Section: Introductionmentioning

confidence: 99%

Numerical methods in phase-change problems

Idelsohn

Storti

Crivelli

1994

ARCO

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This paper summarizes the state of the art of the numerical solution of phase-change problems. After describing the governing equations, a review of the existing methods is presented. The emphasis is put mainly on fixed domain techniques, but a brief description of the main front-tracking methods is included. A special section is devoted to the Newton-Raphson resolution with quadratic convergence of the non-linear system of equations.

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The numerical solution of Stefan problems with front-tracking and smoothing methods

Cited by 14 publications

References 20 publications

The method of lines for Poisson's equation with nonlinear or free boundary conditions

The method of lines for Poisson's equation with nonlinear or free boundary conditions

A flexible framework for simulating and fitting generalized drift-diffusion models

Numerical methods in phase-change problems

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