scholarly journals Rothe time-discretization method for a nonlocal problem arising in thermoelasticity

2005 ◽  
Vol 2005 (1) ◽  
pp. 13-28 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Continuous Dependence ◽  
Nonlocal Problem ◽  
Time Discretization ◽  
Boundary Integral ◽  
Time Variable ◽  
Discretization Method ◽  
Integral Conditions ◽  
Rothe Method ◽  
Semidiscrete Approximation ◽  
The Given

We investigate a model parabolic mixed problem with purely boundary integral conditions arising in the context of thermoelasticity. Using the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable, the questions of existence, uniqueness, and continuous dependence upon data of a weak solution are proved. Moreover, we establish convergence and derive an error estimate for a semidiscrete approximation.

scholarly journals Rothe time-discretization method for the semilinear heat equation subject to a nonlocal boundary condition

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Time Discretization ◽  
Neumann Condition ◽  
Integral Boundary ◽  
Semilinear Heat Equation ◽  
Rothe Method ◽  
The Given

This paper is devoted to prove, in a nonclassical function space, the weak solvability of a mixed problem which combines a Neumann condition and an integral boundary condition for the semilinear one-dimensional heat equation. The investigation is made by means of approximation by the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable.

scholarly journals Rothe method for a mixed problem with an integral condition for the two-dimensional diffusion equation

2003 ◽  
Vol 2003 (16) ◽  
pp. 899-922 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Diffusion Equation ◽  
Integral Condition ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Dimensional Diffusion ◽  
Rothe Method ◽  
Semidiscrete Approximation ◽  
Initial Boundary Value ◽  
The Given

This paper deals with an initial boundary value problem with an integral condition for the two-dimensional diffusion equation. Thanks to an appropriate transformation, the study of the given problem is reduced to that of a one-dimensional problem. Existence, uniqueness, and continuous dependence upon data of a weak solution of this latter are proved by means of the Rothe method. Besides, convergence and an error estimate for a semidiscrete approximation are obtained.

scholarly journals Solution of a transmission problem for semilinear parabolic-hyperbolic equations by the time-discretization method

2006 ◽  
Vol 2006 ◽  
pp. 1-23 ◽  
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Continuous Dependence ◽  
Hyperbolic Equations ◽  
Time Discretization ◽  
Transmission Problem ◽  
Discretization Method ◽  
Convergence Results ◽  
Semilinear Parabolic

We consider a transmission problem for semilinear parabolic-hyperbolic equations. Existence, uniqueness, and continuous dependence of the solution upon the data are proved by using the time-discretization method. Besides, some convergence results of the approximations are established.

Boundary Integral Equation Formulation in Generalized Linear Thermo-Viscoelasticity With Rheological Volume

2003 ◽  
Vol 70 (5) ◽  
pp. 661-667 ◽  
Author(s):  
A. S. El-Karamany
Keyword(s):  
Integral Equation ◽  
Rheological Properties ◽  
Boundary Integral Equation ◽  
Relaxation Times ◽  
Fundamental Solutions ◽  
Boundary Integral ◽  
Phase Lag ◽  
Integral Equation Formulation ◽  
Special Cases ◽  
The Given

A general model of generalized linear thermo-viscoelasticity for isotropic material is established taking into consideration the rheological properties of the volume. The given model is applicable to three generalized theories of thermoelasticity: the generalized theory with one (Lord-Shulman theory) or with two relaxation times (Green-Lindsay theory) and with dual phase-lag (Chandrasekharaiah-Tzou theory) as well as to the dynamic coupled theory. The cases of thermo-viscoelasticity of Kelvin-Voigt model or thermoviscoelasticity ignoring the rheological properties of the volume can be obtained from the given model. The equations of the corresponding thermoelasticity theories result from the given model as special cases. A formulation of the boundary integral equation (BIE) method, fundamental solutions of the corresponding differential equations are obtained and an example illustrating the BIE formulation is given.

Simulation of Water Droplet Merging under Shock Wave Using Real Ghost Fluid Method

2013 ◽  
Vol 444-445 ◽  
pp. 628-632
Author(s):  
Ru Chao Shi ◽  
Sheng Li Xu ◽  
Ya Jun Zhang
Keyword(s):  
Shock Wave ◽  
Interpolation Method ◽  
Time Discretization ◽  
Problem Solver ◽  
Ghost Fluid Method ◽  
Numerical Accuracy ◽  
Air Water Interface ◽  
Fifth Order ◽  
The Given ◽  
Flow States

This paper presents a 3D numerical simulation of water droplets merging under a given shock wave. We couple interpolation method to RGFM (Real Ghost Fluid Method) to improve the numerical accuracy of RGFM. The flow states of air-water interface are calculated by ARPS (approximate Riemann problem solver). Flow field is solved by Euler equation with fifth-order WENO spatial discretization and fourth-order R-K (Runge-Kutta) time discretization. We also employ fifth-order HJ-WENO to discretize level set equation to keep track of gas-liquid interface. Numerical results demonstrate that droplets shape has little change before merging and the merged droplet gradually becomes umbrella-shaped under the given shock wave. We verify that combination of RGFM with interpolation method has the property of reducing numerical error by comparing to the results without employment of interpolation method.

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