scholarly journals Conservative finite volume element schemes for the complex modified Korteweg–de Vries equation

2017 ◽  
Vol 27 (3) ◽  
pp. 515-525 ◽  
Author(s):  
Jin-Liang Yan ◽  
Liang-Hong Zheng
Keyword(s):  
Finite Volume ◽  
Vries Equation ◽  
Numerical Study ◽  
Volume Element ◽  
Element Approximation ◽  
Element Scheme ◽  
De Vries ◽  
Long Time ◽  
Discrete Variational Derivative Method ◽  
Finite Volume Element

AbstractThe aim of this paper is to build and validate a class of energy-preserving schemes for simulating a complex modified Korteweg–de Vries equation. The method is based on a combination of a discrete variational derivative method in time and finite volume element approximation in space. The resulting scheme is accurate, robust and energy-preserving. In addition, for comparison, we also develop a momentum-preserving finite volume element scheme and an implicit midpoint finite volume element scheme. Finally, a complete numerical study is developed to investigate the accuracy, conservation properties and long time behaviors of the energy-preserving scheme, in comparison with the momentum-preserving scheme and the implicit midpoint scheme, for the complex modified Korteweg–de Vries equation.

New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation

2017 ◽  
Vol 9 (2) ◽  
pp. 250-271 ◽  
Author(s):  
Jinliang Yan ◽  
Ming-Chih Lai ◽  
Zhilin Li ◽  
Zhiyue Zhang
Keyword(s):  
Wave Equation ◽  
Finite Volume ◽  
Total Mass ◽  
Volume Element ◽  
Von Neumann ◽  
Element Scheme ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Discrete Variational Derivative Method ◽  
Finite Volume Element

AbstractIn this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.

scholarly journals A Fully Discrete Symmetric Finite Volume Element Approximation of Nonlocal Reactive Flows in Porous Media

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhe Yin ◽  
Qiang Xu
Keyword(s):  
Porous Media ◽  
Finite Volume ◽  
Volume Element ◽  
Optimal Order ◽  
Element Approximation ◽  
Two Dimensional ◽  
Reactive Flows ◽  
Flows In Porous Media ◽  
Fully Discrete ◽  
Finite Volume Element

We study symmetric finite volume element approximations for two-dimensional parabolic integrodifferential equations, arising in modeling of nonlocal reactive flows in porous media. It is proved that symmetric finite volume element approximations are convergent with optimal order inL2-norm. Numerical example is presented to illustrate the accuracy of our method.

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