scholarly journals Análisis de Von Neumann para el métodoLocal Discontinuous Galerkin en 1D

2019 ◽  
Vol 37 (2) ◽  
pp. 199-217
Author(s):  
Paul Castillo ◽  
Sergio Gómez
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
High Order ◽  
Stability Conditions ◽  
Von Neumann ◽  
Theoretical Tool ◽  
Von Neumann Analysis ◽  
Time Marching ◽  
The Stability ◽  
Theoretical Results

Using the von Neumann analysis as a theoretical tool, an analysisof the stability conditions of some explicit time marching schemes, in com-bination with the spatial discretizationLocal Discontinuous Galerkin(LDG)and high order approximations, is presented. The stabilityconstant, CFL(Courant-Friedrichs-Lewy), is studied as a function of theLDG parametersand the approximation degree. A series of numerical experiments is carriedout to validate the theoretical results.

scholarly journals High-order compact LOD methods for solving high-dimensional advection equations

2021 ◽  
Vol 40 (3) ◽  
Author(s):  
Bo Hou ◽  
Yongbin Ge
Keyword(s):  
Truncation Error ◽  
Three Dimensional ◽  
Taylor Series Expansion ◽  
High Order ◽  
High Dimensional ◽  
Analysis Method ◽  
Order Accuracy ◽  
Von Neumann ◽  
One Dimensional ◽  
Von Neumann Analysis

AbstractIn this paper, by using the local one-dimensional (LOD) method, Taylor series expansion and correction for the third derivatives in the truncation error remainder, two high-order compact LOD schemes are established for solving the two- and three- dimensional advection equations, respectively. They have the fourth-order accuracy in both time and space. By the von Neumann analysis method, it shows that the two schemes are unconditionally stable. Besides, the consistency and convergence of them are also proved. Finally, numerical experiments are given to confirm the accuracy and efficiency of the present schemes.

scholarly journals On the exact and numerical solutions to a nonlinear model arising in mathematical biology

2018 ◽  
Vol 22 ◽  
pp. 01061 ◽  
Author(s):  
Asif Yokus ◽  
Tukur Abdulkadir Sulaiman ◽  
Haci Mehmet Baskonus ◽  
Sibel Pasali Atmaca
Keyword(s):  
Analytical Solutions ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Hyperbolic Function ◽  
Von Neumann ◽  
Convection Diffusion Equation ◽  
Wolfram Mathematica ◽  
Forward Difference ◽  
Von Neumann Analysis ◽  
The Stability

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ(η))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L2 and L∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

ATTRACTABILITY AND LOCATION OF EQUILIBRIUM POINT OF CELLULAR NEURAL NETWORKS WITH TIME-VARYING DELAYS

2004 ◽  
Vol 14 (05) ◽  
pp. 337-345 ◽  
Author(s):  
ZHIGANG ZENG ◽  
DE-SHUANG HUANG ◽  
ZENGFU WANG
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Equilibrium Point ◽  
Global Exponential Stability ◽  
Cellular Neural Networks ◽  
Stability Conditions ◽  
Time Varying ◽  
The Stability ◽  
Theoretical Results

This paper presents new theoretical results on global exponential stability of cellular neural networks with time-varying delays. The stability conditions depend on external inputs, connection weights and delays of cellular neural networks. Using these results, global exponential stability of cellular neural networks can be derived, and the estimate for location of equilibrium point can also be obtained. Finally, the simulating results demonstrate the validity and feasibility of our proposed approach.

scholarly journals Collocation for High-Order Differential Equations with Lidstone Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Francesco Costabile ◽  
Anna Napoli
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Numerical Solution ◽  
Collocation Method ◽  
Numerical Experiments ◽  
Recurrence Formula ◽  
High Order ◽  
Theoretical Results

A class of methods for the numerical solution of high-order differential equations with Lidstone and complementary Lidstone boundary conditions are presented. It is a collocation method which provides globally continuous differentiable solutions. Computation of the integrals which appear in the coefficients is generated by a recurrence formula. Numerical experiments support theoretical results.

Insights from von Neumann analysis of high-order flux reconstruction schemes

2011 ◽  
Vol 230 (22) ◽  
pp. 8134-8154 ◽  
Author(s):  
P.E. Vincent ◽  
P. Castonguay ◽  
A. Jameson
Keyword(s):  
High Order ◽  
Flux Reconstruction ◽  
Von Neumann ◽  
Von Neumann Analysis

scholarly journals Preconditioning Techniques Based on the Birkhoff–von Neumann Decomposition

2017 ◽  
Vol 17 (2) ◽  
pp. 201-215 ◽  
Author(s):  
Michele Benzi ◽  
Bora Uçar
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Sparse Matrices ◽  
Stochastic Matrices ◽  
Preconditioning Techniques ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Theoretical Results

AbstractWe introduce a class of preconditioners for general sparse matrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

scholarly journals Stability of Exact and Discrete Energy for Non-Fickian Reaction-Diffusion Equations with a Variable Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Dongfang Li ◽  
Chao Tong ◽  
Jinming Wen
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Variable Delay ◽  
Discrete Energy ◽  
Continuous Problems ◽  
Rate Of Decay ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

scholarly journals High order algorithms for numerical solution of fractional differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammad Shahbazi Asl ◽  
Mohammad Javidi ◽  
Yubin Yan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Algorithms ◽  
Interpolation Polynomial ◽  
High Order ◽  
The Novel ◽  
Fractional Differential ◽  
Novel Algorithms ◽  
The Stability ◽  
Theoretical Results

AbstractIn this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.

scholarly journals Well-Balanced High-Order Discontinuous Galerkin Methods for Systems of Balance Laws

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 15
Author(s):  
Ernesto Guerrero Fernández ◽  
Cipriano Escalante ◽  
Manuel J. Castro Díaz
Keyword(s):  
Numerical Method ◽  
Discontinuous Galerkin ◽  
Stationary Solutions ◽  
High Order ◽  
General Strategy ◽  
Balance Laws ◽  
Smooth Solutions ◽  
Systems Of Balance Laws ◽  
Spurious Oscillations ◽  
Time Marching

This work introduces a general strategy to develop well-balanced high-order Discontinuous Galerkin (DG) numerical schemes for systems of balance laws. The essence of our approach is a local projection step that guarantees the exactly well-balanced character of the resulting numerical method for smooth stationary solutions. The strategy can be adapted to some well-known different time marching DG discretisations. Particularly, in this article, Runge–Kutta DG and ADER DG methods are studied. Additionally, a limiting procedure based on a modified WENO approach is described to deal with the spurious oscillations generated in the presence of non-smooth solutions, keeping the well-balanced properties of the scheme intact. The resulting numerical method is then exactly well-balanced and high-order in space and time for smooth solutions. Finally, some numerical results are depicted using different systems of balance laws to show the performance of the introduced numerical strategy.

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