scholarly journals The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ziwen Jiang ◽  
Deren Xie
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Optimal Order ◽  
Longitudinal Motion ◽  
Order Error ◽  
Fully Discrete ◽  
Backward Euler ◽  
Volume Method

We present a linear backward Euler fully discrete finite volume method for the initial-boundary-value problem of purely longitudinal motion of a homogeneous bar and an give optimal order error estimates inL2andH1norms. Furthermore, we obtain the superconvergence error estimate of the generalized projection of the solutionuinH1norm. Numerical experiment illustrates the convergence and stability of this scheme.

Analysis of a New Mixed Finite Volume Method

2003 ◽  
Vol 3 (1) ◽  
pp. 189-201 ◽  
Author(s):  
Ilya D. Mishev
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Elliptic Equations ◽  
Variable Pressure ◽  
Order Error ◽  
First Order ◽  
Scalar Variable ◽  
Volume Method ◽  
Well Posed ◽  
Theoretical Results

AbstractA new mixed finite volume method for elliptic equations with tensor coefficients on rectangular meshes (2 and 3-D) is presented. The implementation of the discretization as a finite volume method for the scalar variable (“pressure”) is derived. The scheme is well suited for heterogeneous and anisotropic media because of the generalized harmonic averaging. It is shown that the method is stable and well posed. First-order error estimates are derived. The theoretical results are confirmed by the presented numerical experiments.

Stable Non-symmetric Coupling of the Finite Volume Method and the Boundary Element Method for Convection-Dominated Parabolic-Elliptic Interface Problems

2020 ◽  
Vol 20 (2) ◽  
pp. 251-272
Author(s):  
Christoph Erath ◽  
Robert Schorr
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Time Discretization ◽  
Euler Method ◽  
Interface Problems ◽  
Backward Euler Method ◽  
Elliptic Interface Problems ◽  
Practical Applications ◽  
Backward Euler ◽  
Volume Method

AbstractMany problems in electrical engineering or fluid mechanics can be modeled by parabolic-elliptic interface problems, where the domain for the exterior elliptic problem might be unbounded. A possibility to solve this class of problems numerically is the non-symmetric coupling of finite elements (FEM) and boundary elements (BEM) analyzed in [H. Egger, C. Erath and R. Schorr, On the nonsymmetric coupling method for parabolic-elliptic interface problems, SIAM J. Numer. Anal. 56 2018, 6, 3510–3533]. If, for example, the interior problem represents a fluid, this method is not appropriate since FEM in general lacks conservation of numerical fluxes and in case of convection dominance also stability. A possible remedy to guarantee both is the use of the vertex-centered finite volume method (FVM) with an upwind stabilization option. Thus, we propose a (non-symmetric) coupling of FVM and BEM for a semi-discretization of the underlying problem. For the subsequent time discretization we introduce two options: a variant of the backward Euler method which allows us to develop an analysis under minimal regularity assumptions and the classical backward Euler method. We analyze both, the semi-discrete and the fully-discrete system, in terms of convergence and error estimates. Some numerical examples illustrate the theoretical findings and give some ideas for practical applications.

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 Error Estimates for the Heterogeneous Multiscale Finite Volume Method of Convection-Diffusion-Reaction Problem

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-6
Author(s):  
Tao Yu ◽  
Peichang Ouyang ◽  
Haitao Cao
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Diffusion Reaction ◽  
Optimal Order ◽  
Convection Diffusion ◽  
Heterogeneous Multiscale ◽  
Multiscale Finite Volume ◽  
Volume Method ◽  
Optimal Order Convergence ◽  
Reaction Problem

Based on the heterogeneous multiscale method, this paper presents a finite volume method to solve multiscale convection-diffusion-reaction problem. The paper constructs an algorithm of the optimal order convergence rate in H1-norm under periodic medias.

Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow

2014 ◽  
Vol 6 (5) ◽  
pp. 663-679
Author(s):  
Jianhong Yang ◽  
Gang Lei ◽  
Jianwei Yang
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Incompressible Flow ◽  
Stokes Equations ◽  
Practical Importance ◽  
Optimal Order ◽  
Fine Mesh ◽  
Non Linear System ◽  
Volume Method ◽  
Mesh Convergence

AbstractIn this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pairP1—P1which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in theH1-norm for velocity and theL2-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relationh = 𝾪(H2). Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

scholarly journals Discontinuous Mixed Covolume Methods for Parabolic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ailing Zhu ◽  
Ziwen Jiang
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Error Estimates ◽  
Parabolic Problems ◽  
Optimal Order ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Optimal Order Error Estimates ◽  
Backward Euler

We present the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume schemes for parabolic problems on triangular meshes. We give the error analysis of the discontinuous mixed covolume schemes and obtain optimal order error estimates in discontinuousHdivand first-order error estimate inL2.

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