Comparative study of six explicit and two implicit finite difference schemes for solving one‐dimensional parabolic partial differential equations

Roberts, David L.; Selim, M. Sami

doi:10.1002/nme.1620200504

Cited by 18 publications

(5 citation statements)

References 10 publications

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“…The comparison between six explicit (such as the stable explicit method) and two implicit methods from accuracy and performance viewpoints was performed by Roberts and Selim. [83] Thibault [84] investigated nine commonly known three-dimensional finite difference methods for solving the heat diffusion equation. He demonstrated that the Euler implicit and the CrankNicolson methods are prohibitively demanding in computer time.…”

Section: Methodsmentioning

confidence: 99%

Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems

Tavakoli

Davami

2007

Metall Mater Trans B

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An unconditionally stable fully explicit finite difference method for solution of conduction dominated phase-change problems is presented. This method is based on an asymmetric stable finite difference scheme for approximation of diffusion terms and application of the temperature recovery method as a phase-change modeling method. The computational cost of the presented method is the same as the explicit method per time-step, while it is free from time-step limitation due to stability criteria. It robustly handles isothermal and nonisothermal phase-change problems and is very efficient when the global temperature field is desirable (not accurate front position). The robustness, stability, accuracy, and efficiency of the presented method are demonstrated with several benchmarks. Comparison with some stable implicit time integration methods is also included. Numerical experiments show that the evaluated temperature field for large values of the Fourier numbers has good/reasonable agreement with the result of small Fourier numbers and the exact/reference solution.

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Section: Methodsmentioning

confidence: 99%

Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems

Tavakoli

Davami

2007

Metall Mater Trans B

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“…In general, the numerical solution of the governing PDE for flood wave propagation is derived by means of explicit and implicit numerical schemes, such as the finite difference, element, and volume methods. Roberts and Selim indicated that explicit and implicit schemes each have their own advantages. The selection of an appropriate numerical scheme depends on accuracy, computation time, and programming effort.…”

Section: Methodsmentioning

confidence: 99%

Modeling of uncertainty for flood wave propagation induced by variations in initial and boundary conditions using expectation operator on explicit numerical solutions

Hsu

2017

Numerical Meth Engineering

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This study proposes a framework for uncertainty analysis by incorporating explicit numerical solutions of governing equations for flood wave propagation with the expectation operator. It aims at effectively evaluating the effect of variations in initial and boundary conditions on the estimation of flood waves. Spatiotemporal semivariogram models are employed to quantify the correlation of the variables in time and space. The 1D nonlinear kinematic wave equation for the overland flow (named EVO_NS_KWE) is applied in the model development. Model validation is made by comparison with the Monte Carlo simulation model in the calculation of statistical properties of model outputs (ie, flow depths), that is, the mean, standard deviation, and coefficient of variation. The results from the model validation show that the EVO_NS_KWE model can produce excellent approximations of the mean and less satisfactory approximations of the standard deviation and coefficient of variation compared with those obtained by using the Monte Carlo simulation model. It concludes that the uncertainties of flow depths in the domain are significantly affected by variations in the boundary condition. Future application of the EVO_NS_KWE model enables the evaluation of uncertainty in model outputs induced by the initial and boundary condition subject to uncertainty and will also provide corresponding probabilistic information for risk quantification method.

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“…Numerical approximation of the solution was obtained by a finite difference discretization of (2) using a parallelized FTCS explicit method. This generally used scheme provides a good trade-off between accuracy and programming effort [14].…”

Section: (I)mentioning

confidence: 99%

Simulating Solid Tumors with a Microenvironment-Coupled Agent-Based Computational Model

Kiss

Lovrics

2018

Acta Universitatis Sapientiae Electrical and Mechanical Engineering

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In this paper, we introduce a three-dimensional lattice-based computational model in which every lattice point can be occupied by an agent of various types (e.g. cancer cell, blood vessel cell or extracellular matrix). The behavior of agents can be associated to different chemical compounds that obey mass-transfer laws such as diffusion and decay in the surrounding environment. Furthermore, agents are also able to produce and consume chemical compounds. After a detailed description, the capabilities of the model are demonstrated by presenting and discussing a simulation of a biological experiment available in the literature.

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Comparative study of six explicit and two implicit finite difference schemes for solving one‐dimensional parabolic partial differential equations

Cited by 18 publications

References 10 publications

Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems

Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems

Modeling of uncertainty for flood wave propagation induced by variations in initial and boundary conditions using expectation operator on explicit numerical solutions

Simulating Solid Tumors with a Microenvironment-Coupled Agent-Based Computational Model

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