scholarly journals Two-layer finite-difference schemes for the Korteweg-de Vries equation in Euler variables

2020 ◽  
Vol 49 ◽  
pp. 57-69
Author(s):  
Vladimir Ivanovich Mazhukin ◽  
◽  
Aleksandr Viktorovich Shapranov ◽  
Elena Nikolaevna Bykovskaya
Keyword(s):  
Finite Difference ◽  
Vries Equation ◽  
Approximation Error ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Family ◽  
Implicit Difference Schemes ◽  
Solution Accuracy ◽  
Modified Equation

A family of weighted two-layer finite-difference schemes is presented. Using the example of the numerical solution of model problems on the propagation of a single soliton and the interaction of two solitons, the high quality of explicit-implicit schemes of the Crank-Nichols type with the parameter σ = 0.5 and the order of approximation O(Δt2 + Δx2) isshown. Completely implicit two-layer difference schemes with the parameter σ = 1 and O (Δt+ Δx2) are characterized by absolute stability with a low solution accuracy due to a highapproximation error. The family of explicitly implicit difference schemes is absolutely unstable if the explicitness parameter σ <0.5 prevails. Analysis of the structure of the approximation error, performed using the modified equation method, confirmed the results of numerical simulation.

On Finite Difference Schemes for Elliptic Equations with Discontinuous Coefficients

2013 ◽  
Vol 13 (3) ◽  
pp. 281-289
Author(s):  
Manfred Dobrowolski
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Elliptic Equations ◽  
Approximation Error ◽  
Discontinuous Coefficients ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
The Finite Element Method ◽  
Order Four ◽  
Element Theory

Abstract. We study the convergence of finite difference schemes for approximating elliptic equations of second order with discontinuous coefficients. Two of these finite difference schemes arise from the discretization by the finite element method using bilinear shape functions. We prove an convergence for the gradient, if the solution is locally in H3. Thus, in contrast to the first order convergence for the gradient obtained by the finite element theory we show that the gradient is superclose. From the Bramble–Hilbert Lemma we derive a higher order compact (HOC) difference scheme that gives an approximation error of order four for the gradient. A numerical example is given.

Finite-difference schemes for the korteweg—de vries equation

2000 ◽  
Vol 36 (5) ◽  
pp. 789-797 ◽  
Author(s):  
V. I. Mazhukin ◽  
P. P. Matus ◽  
I. A. Mikhailyuk
Keyword(s):  
Finite Difference ◽  
Vries Equation ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
De Vries

scholarly journals Iterative Methods for Obtaining Energy-Minimizing Parametric Snakes with Applications to Medical Imaging

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Alexandru Ioan Mitrea ◽  
Radu Badea ◽  
Delia Mitrea ◽  
Sergiu Nedevschi ◽  
Paulina Mitrea ◽  
...  
Keyword(s):  
Medical Imaging ◽  
Numerical Methods ◽  
Finite Difference ◽  
Iterative Method ◽  
Iterative Methods ◽  
Approximation Error ◽  
Deformable Models ◽  
Difference Schemes ◽  
Finite Difference Schemes

After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical methods that implement the above numerical methods are also pointed out.

scholarly journals MODIFIED EQUATION FOR A CLASS OF EXPLICIT AND IMPLICIT SCHEMES SOLVING ONE-DIMENSIONAL ADVECTION PROBLEM

2021 ◽  
Vol 61 (SI) ◽  
pp. 49-58
Author(s):  
Tomáš Bodnár ◽  
Philippe Fraunié ◽  
Karel Kozel
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Advection Equation ◽  
One Dimensional ◽  
Implicit Schemes ◽  
Explicit And Implicit Schemes ◽  
Spatial Support ◽  
Modified Equation

This paper presents the general modified equation for a family of finite-difference schemes solving one-dimensional advection equation. The whole family of explicit and implicit schemes working at two time-levels and having three point spatial support is considered. Some of the classical schemes (upwind, Lax-Friedrichs, Lax-Wendroff) are discussed as examples, showing the possible implications arising from the modified equation to the properties of the considered numerical methods.

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