scholarly journals A Conservative and Implicit Second-Order Nonlinear Numerical Scheme for the Rosenau-KdV Equation

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1183
Author(s):  
Cui Guo ◽  
Yinglin Wang ◽  
Yuesheng Luo
Keyword(s):  
Differential Equation ◽  
Finite Difference ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Numerical Scheme ◽  
Kdv Equation ◽  
Multiple Integral ◽  
Numerical Results ◽  
Multiple Integration ◽  
Volume Method

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is O(τ2+h3). In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective.

scholarly journals Pixels and their neighbors: Finite volume

2016 ◽  
Vol 35 (8) ◽  
pp. 703-706 ◽  
Author(s):  
Rowan Cockett ◽  
Lindsey J. Heagy ◽  
Douglas W. Oldenburg
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Direct Current ◽  
Numerical Results ◽  
Dc Resistivity ◽  
Matrix Representations ◽  
Volume Method

We take you on the journey from continuous equations to their discrete matrix representations using the finite-volume method for the direct current (DC) resistivity problem. These techniques are widely applicable across geophysical simulation types and have their parallels in finite element and finite difference. We show derivations visually, as you would on a whiteboard, and have provided an accompanying notebook at http://github.com/seg to explore the numerical results using SimPEG ( Cockett et al., 2015 ).

scholarly journals A conservative numerical scheme for Rosenau-RLW equation based on multiple integral finite volume method

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Cui Guo ◽  
Fang Li ◽  
Wenping Zhang ◽  
Yuesheng Luo
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Lagrange Interpolation ◽  
Numerical Scheme ◽  
Multiple Integral ◽  
Third Order ◽  
Unconditionally Stable ◽  
Fully Discrete ◽  
Second Order In Time ◽  
Volume Method

Abstract A multiple integral finite volume method combined and Lagrange interpolation are applied in this paper to the Rosenau-RLW (RRLW) equation. We construct a two-level implicit fully discrete scheme for the RRLW equation. The numerical scheme has the accuracy of third order in space and second order in time, respectively. The solvability and uniqueness of the numerical solution are shown. We verify that the numerical scheme keeps the original equation characteristic of energy conservation. It is proved that the numerical scheme is convergent in the order of $O(\tau ^{2} + h^{3})$ O ( τ 2 + h 3 ) and unconditionally stable. A numerical experiment is given to demonstrate the validity and accuracy of scheme.

scholarly journals SIMPLE ALGORITHMS FOR CALCULATION OF THE AXIAL‐SYMMETRIC HEAT TRANSPORT PROBLEM IN A CYLINDER

2001 ◽  
Vol 6 (2) ◽  
pp. 262-269 ◽  
Author(s):  
H. Kalis ◽  
A. Lasis
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Scheme ◽  
Ordinary Differential Equations ◽  
Finite Volume Method ◽  
Heat Transport ◽  
Finite Volume ◽  
Transport Problem ◽  
Quadrature Formulae ◽  
Volume Method

The approximation of axial‐symmetric heat transport problem in a cylinder is based on the finite volume method. In the classical formulation of the finite volume method it is assumed that the flux terms in the control volume are approximated with the finite difference expressions. Then in the 1‐D case the corresponding finite difference scheme for the given source function is not exact. There we propose the exact difference scheme. In 2‐D case the corresponding integrals are approximated using different quadrature formulae. This procedure allows one to reduce the heat transport problem described by a partial differential equation to an initial‐value problem for a system of two ordinary differential equations of the order depending on the quadrature formulae used. Numerical solutions of the corresponding algorithms are obtained using MAPLE routines for stiff system of ordinary differential equations.

Heat Conduction Equation Finite Volume Method to Achieve on MATLAB

2012 ◽  
Vol 195-196 ◽  
pp. 712-717
Author(s):  
Qiong Xue ◽  
Xiao Feng Xiao ◽  
Niang Zhi Fan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Heat Conduction Equation ◽  
Conduction Equation ◽  
Two Dimensional ◽  
Partial Differential ◽  
Volume Method

Diffusion only, two dimensional heat conduction has been described on partial differential equation. Based on Finite Volume Method, Discretized algebraic Equation of partial differential equation have been deduced. different coefficients and source terms have been discussed under different boundary conditions, which include prescribed heat flux, prescribed temperature, convection and insulated. Transient heat conduction analysises of infinite plate with uniform thickness and two dimensional rectangle region have been realized by programming using MATLAB. It is useful to make the heat conduction equation more understandable by its solution with graphical expression, feasibility and stability of numerical method have been demonstrated by running result.

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