Topic
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.
Papers published on a yearly basis
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Papers
TL;DR: In this article, numerical methods for solving the fractional-in-space Allen-Cahn equation with small perturbation parameters and strong nonlinearity were considered, and the numerical solutions satisfy discrete maximum principle under reasonable time step constraint.
Abstract: We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only $$\mathcal {O}(N\log N)$$
computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.
161 citations
TL;DR: In this paper, a numerical simulation is carried out by solving the governing continuity, momentum and energy equations for laminar flow in curvilinear coordinates using the Finite Difference (FD) approach.
161 citations
TL;DR: In this article, a numerical study of the three-dimensional fluid dynamics inside a model left ventricle during diastole is presented, which is modelled as a portion of a prolate spheroid with a moving wall, whose dynamics is externally forced to agree with a simplified waveform of the entering flow.
Abstract: A numerical study of the three-dimensional fluid dynamics inside a model left ventricle during diastole is presented. The ventricle is modelled as a portion of a prolate spheroid with a moving wall, whose dynamics is externally forced to agree with a simplified waveform of the entering flow. The flow equations are written in the meridian body-fitted system of coordinates, and expanded in the azimuthal direction using the Fourier representation. The harmonics of the dependent variables are normalized in such a way that they automatically satisfy the high-order regularity conditions of the solution at the singular axis of the system of coordinates. The resulting equations are solved numerically using a mixed spectral–finite differences technique. The flow dynamics is analysed by varying the governing parameters, in order to understand the main fluid phenomena in an expanding ventricle, and to obtain some insight into the physiological pattern commonly detected. The flow is characterized by a well-defined structure of vorticity that is found to be the same for all values of the parameters, until, at low values of the Strouhal number, the flow develops weak turbulence.
161 citations
TL;DR: Numerical tests revealed that if the number of points selected by DEIM algorithm reached 50, the approximation errors due to POD/DEIM and POD reduced systems have the same orders of magnitude, thus supporting the theoretical results existing in the literature.
161 citations
TL;DR: A multilevel approach for the solution of partial differential equations based on a multiscale basis which is constructed from a one-dimensional multiscales basis by the tensor product approach, which is well suited for higher dimensional problems.
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
160 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 53 |
| 2024 | 136 |
| 2023 | 170 |
| 2022 | 357 |
| 2021 | 733 |
| 2020 | 690 |