Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

Artificial Neural Network Method for Solution of Boundary Value Problems With Exact Satisfaction of Arbitrary Boundary Conditions

Abstract: A method for solving boundary value problems (BVPs) is introduced using artificial neural networks (ANNs) for irregular domain boundaries with mixed Dirichlet/Neumann boundary conditions (BCs). The approximate ANN solution automatically satisfies BCs at all stages of training, including before training commences. This method is simpler than other ANN methods for solving BVPs due to its unconstrained nature and because automatic satisfaction of Dirichlet BCs provides a good starting approximate solution for significant portions of the domain. Automatic satisfaction of BCs is accomplished by the introduction of an innovative length factor. Several examples of BVP solution are presented for both linear and nonlinear differential equations in two and three dimensions. Error norms in the approximate solution on the order of 10-4 to 10-5 are reported for all example problems.

Free Convection in a Square Cavity Filled with a Porous Medium Saturated by Nanofluid Using Tiwari and Das’ Nanofluid Model

Abstract: Free convection in a square differentially heated porous cavity filled with a nanofluid is numerically investigated. The mathematical model has been formulated in dimensionless stream function and temperature taking into account the Darcy–Boussinesq approximation. The Tiwari and Das’ nanofluid model with new more realistic empirical correlations for the physical properties of the nanofluids has been used for numerical analysis. The governing equations have been solved numerically on the basis of a second-order accurate finite difference method. The developed algorithm has been validated by direct comparisons with previously published papers and the results have been found to be in good agreement. The results have been presented in terms of the streamlines, isotherms, local, and average Nusselt numbers at left vertical wall at a wide range of key parameters.

Full-vectorial mode calculations by finite difference method

Abstract: Finite difference calculations of full-vectorial modes of optical waveguides are presented. This method has overcome the limitations of the semivectorial approximation and is able to calculate full-vectorial modes of arbitrary order for a given structure with an arbitrary refractive index profile. Numerical results show that the method is accurate. In addition, the leakage phenomenon of the higher modes of a rib waveguide is predicted by this method.

Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences

Abstract: We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a one-dimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments. — Author's Abstract