A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

Crank, John P.; Nicolson, P.

doi:10.1007/bf02127704

“…The resulting equations for velocity components are nonlinear, and are solved using the NewtonRaphson iterative algorithm [16][17][18]. The diffusion step equations: (7) are discretized using finite differences with the Crank-Nicolson time-integration scheme [19]. Equations (7) are linearized with the alternating direction implicit (ADI) method [20] resulting in a three-diagonal system of equations.…”

Section: Mobed2 Codementioning

confidence: 99%

Different approaches to two-dimensional numerical modelling of natural watercourses

2018

JCE

1

0

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Different approaches to two-dimensional numerical modelling of natural watercoursesResults obtained by analysis of three different approaches to numerical modelling of open watercourses are presented in the paper. Two examples of flow, uniform flow on the physical model of a curve at the Tisza River and nonuniform flow at the Danube River, are analysed. Standard error analysis has revealed that results of the similar order of magnitude are obtained in implementation of two numerical methods in which computational grids are used, while the method in which computational grid is not used, i.e. the smoothed particle hydrodynamics method, requires further improvements to enable successful modelling of the gradually variable flow. Key words:2D flow, finite element method, finite difference method, smoothed particle hydrodynamics method Pregledni rad Zoltan Horvat, Mirjana Horvat, Nikola Rosić, Budo Zindović, Radomir Kapor Različiti pristupi dvodimenzionalnom numeričkom modeliranju prirodnih vodotoka U radu su prikazani rezultati analize tri različita pristupa numeričkom modeliranju otvorenih vodotoka. Analizirana su dva primjera strujanja, jednoliko strujanje na fizikalnom modelu krivine na rijeci Tisi i nejednoliko strujanje na rijeci Dunavu. Analiza standardne pogreške pokazala je da se upotrebom dviju numeričkih metoda u kojima se koriste računske mreže dobivaju rezultati istog reda točnosti, dok metoda u kojoj se ne koristi računska mreža, metoda hidrodinamike izglađenih čestica, zahtijeva daljnja poboljšanja kako bi se uspješno modeliralo postupno promjenjivo strujanje.

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“…This article proposes a finite difference method to numerically solve the partial differential equations developed by [7]. The result is …”

Section: Polynomial Chaos Based On the Penkfmentioning

confidence: 99%

Polynomial Chaos based on the parallelized ensemble Kalman filter to estimate precipitation states

Sánchez

¹

,

Infante²,

Marcano³

et al. 2015

Stat., optim. inf. comput.

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This article develops a methodology combining methods of numerical analysis and stochastic differential equations with computational algorithms to treat problems which have complex nonlinear dynamics in high dimensions. A method to estimate parameters and states of a dynamic system is proposed inspired by the parallelized ensemble Kalman filter (PEnKF) and the polynomial chaos theory of Wiener-Askey. The main advantage of the proposal is in providing a precise efficient algorithm with low computational cost. For the analysed data, the methods provide good predictions, spatially and temporally, for the unknown precipitation states for the first 24 hours. Two goodness of fit measures provide confidence in the quality of the model predictions. The performance of the parallel algorithm, measured by the acceleration and efficiency factors, shows an increase of 7% in speed with respect to the sequential version and is most efficient for P = 2 threads.

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“…To achieve this objective it is necessary to solve a stochastic differential equation in terms of the Fokker-Plank equation using a numerical method of finite differences ( [7]). The solution is expressed using an expansion of the polynomial chaos.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Chaos based on the parallelized ensemble Kalman filter to estimate precipitation states

Sánchez

¹

,

Infante²,

Marcano³

et al. 2015

Stat., optim. inf. comput.

0

View full text Add to dashboard Cite

This article develops a methodology combining methods of numerical analysis and stochastic differential equations with computational algorithms to treat problems which have complex nonlinear dynamics in high dimensions. A method to estimate parameters and states of a dynamic system is proposed inspired by the parallelized ensemble Kalman filter (PEnKF) and the polynomial chaos theory of Wiener-Askey. The main advantage of the proposal is in providing a precise efficient algorithm with low computational cost. For the analysed data, the methods provide good predictions, spatially and temporally, for the unknown precipitation states for the first 24 hours. Two goodness of fit measures provide confidence in the quality of the model predictions. The performance of the parallel algorithm, measured by the acceleration and efficiency factors, shows an increase of 7% in speed with respect to the sequential version and is most efficient for P = 2 threads.

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A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

Cited by 829 publications

References 5 publications

Different approaches to two-dimensional numerical modelling of natural watercourses

Different approaches to two-dimensional numerical modelling of natural watercourses

Polynomial Chaos based on the parallelized ensemble Kalman filter to estimate precipitation states

Polynomial Chaos based on the parallelized ensemble Kalman filter to estimate precipitation states

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