Spiral Tip Identification via Deterministic Learning

2019 ◽  
Vol 29 (03) ◽  
pp. 1950040 ◽  
Author(s):  
Xunde Dong ◽  
Chen Song ◽  
Cong Wang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Numerical Experiments ◽  
Spiral Wave ◽  
Spiral Waves ◽  
Wave Source ◽  
Deterministic Learning ◽  
New Perspective

A spiral tip can be considered as a wave source, i.e. a wave is sent out after the tip rotates one circle. Therefore, the dynamics of the spiral tip is vital to understand the behavior of spiral waves. In this paper, we study the spiral tip dynamics from a new perspective by using deterministic learning. A Barkley model described by partial differential equations (PDEs) is employed to illustrate the method. It is first transformed into a set of ordinary differential equations (ODEs) by using finite difference method. Then, the position states of spiral tip are extracted from the spiral wave generated by the transformed Barkley model by using an isocontour method. Finally, with the recurrent trajectory of spiral tip, its dynamics is accurately identified by using the deterministic learning theory. It is shown that the dynamics underlying the periodic or recurrent trajectory of spiral tips can be accurately identified by using the proposed approach. Numerical experiments are included to demonstrate the effectiveness and feasibility of the proposed method.

scholarly journals A parallel in time/spectral collocation combined with finite difference method for the time fractional differential equations

2021 ◽  
Vol 15 ◽  
pp. 174830262110084
Author(s):  
Xianjuan Li ◽  
Yanhui Su
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Spectral Method ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Spectral Collocation ◽  
Parallel Method ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this article, we consider the numerical solution for the time fractional differential equations (TFDEs). We propose a parallel in time method, combined with a spectral collocation scheme and the finite difference scheme for the TFDEs. The parallel in time method follows the same sprit as the domain decomposition that consists in breaking the domain of computation into subdomains and solving iteratively the sub-problems over each subdomain in a parallel way. Concretely, the iterative scheme falls in the category of the predictor-corrector scheme, where the predictor is solved by finite difference method in a sequential way, while the corrector is solved by computing the difference between spectral collocation and finite difference method in a parallel way. The solution of the iterative method converges to the solution of the spectral method with high accuracy. Some numerical tests are performed to confirm the efficiency of the method in three areas: (i) convergence behaviors with respect to the discretization parameters are tested; (ii) the overall CPU time in parallel machine is compared with that for solving the original problem by spectral method in a single processor; (iii) for the fixed precision, while the parallel elements grow larger, the iteration number of the parallel method always keep constant, which plays the key role in the efficiency of the time parallel method.

scholarly journals CMMSE: A technique for generating adapted discretizations to solve partial differential equations with the generalized finite difference method

2021 ◽  
Author(s):  
Augusto César Ferreira ◽  
Miguel Ureña ◽  
HIGINIO RAMOS
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Meshless Method ◽  
Computational Cost ◽  
Partial Differential

The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strategy allows using a smaller number of points and a lower computational cost to achieve the same accuracy that would be obtained with a regular discretization.

scholarly journals A comparative analysis of methods: mimetics, finite differences and finite elements for 1-dimensional stationary problems

2021 ◽  
Vol 8 (1) ◽  
pp. 1-11
Author(s):  
Abdul Abner Lugo Jiménez ◽  
Guelvis Enrique Mata Díaz ◽  
Bladismir Ruiz
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Initial Conditions ◽  
Stationary Regime ◽  
Heat Transfer Fluid ◽  
Mimetic Finite Difference Method ◽  
Linear Differential

Numerical methods are useful for solving differential equations that model physical problems, for example, heat transfer, fluid dynamics, wave propagation, among others; especially when these cannot be solved by means of exact analysis techniques, since such problems present complex geometries, boundary or initial conditions, or involve non-linear differential equations. Currently, the number of problems that are modeled with partial differential equations are diverse and these must be addressed numerically, so that the results obtained are more in line with reality. In this work, a comparison of the classical numerical methods such as: the finite difference method (FDM) and the finite element method (FEM), with a modern technique of discretization called the mimetic method (MIM), or mimetic finite difference method or compatible method, is approached. With this comparison we try to conclude about the efficiency, order of convergence of these methods. Our analysis is based on a model problem with a one-dimensional boundary value, that is, we will study convection-diffusion equations in a stationary regime, with different variations in the gradient, diffusive coefficient and convective velocity.

scholarly journals Modelling the Flow of Mass and Energy within a Snowpack for Hydrological Forecasting

1983 ◽  
Vol 4 ◽  
pp. 198-203 ◽  
Author(s):  
E. M. Morris
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Snow Melt ◽  
Partial Differential ◽  
Hydrological Forecasting ◽  
Using Data

This paper describes a deterministic, distributed snow-melt model which has been developed for the Systeme Hydrologique Européen (SHE), The model is based on partial differential equations describing the flow of mass and energy through the snow. These equations are solved by an implicit, iterative finite-difference method. The behaviour of the model is investigated using data from a sub-Arctic site in Scotland.

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