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The existence of a solution and a numerical method for the Timoshenko nonlinear wave system

Jemal Peradze (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The initial boundary value problem for a beam is considered in the Timoshenko model. Assuming the analyticity of the initial conditions, it is proved that the problem is solvable throughout the time interval. After that, a numerical algorithm, consisting of three steps, is constructed. The solution is approximated with respect to the spatial and time variables using the Galerkin method and a Crank–Nicholson type scheme. The system of equations obtained by discretization is solved by a version...

The far-field modelling of transonic compressible flows

C. A. Coclici, Ivan L. Sofronov, Wolfgang L. Wendland (2001)

Mathematica Bohemica

We present a method for the construction of artificial far-field boundary conditions for two- and three-dimensional exterior compressible viscous flows in aerodynamics. Since at some distance to the surrounded body (e.g. aeroplane, wing section, etc.) the convective forces are strongly dominant over the viscous ones, the viscosity effects are neglected there and the flow is assumed to be inviscid. Accordingly, we consider two different model zones leading to a decomposition of the original flow...

The fourth order accuracy decomposition scheme for an evolution problem

Zurab Gegechkori, Jemal Rogava, Mikheil Tsiklauri (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

The fourth order accuracy decomposition scheme for an evolution problem

Zurab Gegechkori, Jemal Rogava, Mikheil Tsiklauri (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

The generalized finite volume SUSHI scheme for the discretization of the peaceman model

Mohamed Mandari, Mohamed Rhoudaf, Ouafa Soualhi (2021)

Applications of Mathematics

We demonstrate some a priori estimates of a scheme using stabilization and hybrid interfaces applying to partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require special care while discretizing by a finite volume method SUSHI. Later,...

The gradient superconvergence of the finite volume method for a nonlinear elliptic problem of nonmonotone type

Tie Zhu Zhang, Shu Hua Zhang (2015)

Applications of Mathematics

We study the superconvergence of the finite volume method for a nonlinear elliptic problem using linear trial functions. Under the condition of C -uniform meshes, we first establish a superclose weak estimate for the bilinear form of the finite volume method. Then, we prove that on the mesh point set S , the gradient approximation possesses the superconvergence: max P ∈ S | ( ∇ u - ∇ ¯ u h ) ( P ) | = O ( h 2 ) | ln h | 3 / 2 , where ∇ ¯ denotes the average gradient on elements containing vertex P . Furthermore, by using the interpolation post-processing technique,...

The invertibility of the isoparametric mappings for triangular quadratic Lagrange finite elements

Josef Dalík (2012)

Applications of Mathematics

A reference triangular quadratic Lagrange finite element consists of a right triangle K ^ with unit legs S 1 , S 2 , a local space ℒ ^ of quadratic polynomials on K ^ and of parameters relating the values in the vertices and midpoints of sides of K ^ to every function from ℒ ^ . Any isoparametric triangular quadratic Lagrange finite element is determined by an invertible isoparametric mapping ℱ h = ( F 1 , F 2 ) ∈ ℒ ^ × ℒ ^ . We explicitly describe such invertible isoparametric mappings ℱ h for which the images ℱ h ( S 1 ) , ℱ h ( S 2 ) of the segments S 1 , S 2 are segments,...

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