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.
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TL;DR: In this paper, a nonlinear stability criterion for anisotropic continuum traffic flow model is derived by using a wavefront expansion technique, and the stability criterion is illustrated by numerical results using the finite difference method for two different values of the dimensionless parameter.
Abstract: This paper presents our study of the nonlinear stability of a new anisotropic continuum traffic flow model in which the dimensionless parameter or anisotropic factor controls the non-isotropic character and diffusive influence. In order to establish traffic flow stability criterion or to know the critical parameters that lead, on one hand, to a stable response to perturbations or disturbances or, on the other hand, to an unstable response and therefore to a possible congestion, a nonlinear stability criterion is derived by using a wavefront expansion technique. The stability criterion is illustrated by numerical results using the finite difference method for two different values of anisotropic parameter. It is also been observed that the newly derived stability results are consistent with previously reported results obtained using approximate linearisation methods. Moreover, the stability criterion derived in this paper can provide more refined information from the perspective of the capability to reproduce nonlinear traffic flow behaviors observed in real traffic than previously established methodologies.
93 citations
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TL;DR: In this paper, a novel discrete boundary condition for wide angle parabolic equations (WAPEs) is derived from the fully discretized whole-space problem that is reflection-free and yields an unconditionally stable scheme.
93 citations
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TL;DR: The present method is its speed over a range of problems including both fast and slow transients, its accuracy, its stability and its flexibility, which compares very well with the second-order-method of characteristics and the two-step Lax–Wendroff method.
Abstract: Existing methods for the analysis of transient flows in pipe networks are often geared towards certain types of flows such as gas flows vis-a-vis liquid flows or isothermal flows vis-a-vis non-isothermal flows. Also, simplifying assumptions are often made which introduce inaccuracies when the method is applied outside the domain for which it was originally intended. This paper describes an implicit finite difference method based on the simultaneous pressure correction approach which is valid for both liquid and gas flows, for both isothermal and non-isothermal flows and for both fast and slow transients. The problematic convective acceleration term in the momentum equation, often neglected in other methods, is retained but eliminated by casting the momentum equation in an alternative form. The accuracy and stability of the method, depending on a time-step weighing factor α, are illustrated by analysing fast transients in a pipeline and simple branching network. The proposed method compares very well with the second-order-method of characteristics and the two-step Lax–Wendroff method. The advantages of the present method is its speed over a range of problems including both fast and slow transients, its accuracy, its stability and its flexibility. Copyright © 2001 John Wiley & Sons, Ltd.
93 citations
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TL;DR: In this article, a direct application of the nonstandard methods of Mickens allows the construction of a finite-difference scheme that is dynamically consistent with the differential equations for the Lotka-Volterra system.
93 citations
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TL;DR: In this article, the authors give convergence criteria for general difference schemes for boundary value problems in Lipschitzian regions, and prove convergence for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
Abstract: Convergence proofs for the multi-grid iteration are known for the case of finite element equations and for the case of some difference schemes discretizing boundary value problems in a rectangular region. In the present paper we give criteria of convergence that apply to general difference schemes for boundary value problems in Lipschitzian regions. Furthermore, convergence is proved for the multi-grid algorithm with Gauss-Seidel's iteration as smoothing procedure.
93 citations