The nonlinear modified equation approach to analyzing finite difference schemes

1981 ◽  
Author(s):  
G. KLOPFER ◽  
D. MCRAE
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Equation Approach ◽  
Modified Equation

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.

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.

scholarly journals Conservative finite difference schemes for the modified Camassa-Holm equation

JSIAM Letters ◽  
2011 ◽  
Vol 3 (0) ◽  
pp. 37-40 ◽  
Author(s):  
Yuto Miyatake ◽  
Takayasu Matsuo ◽  
Daisuke Furihata
Keyword(s):  
Finite Difference ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Holm Equation
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