Three-level Finite Difference Scheme Research Articles

A three-level finite difference method with preconditioning technique for two-dimensional nonlinear fractional complex Ginzburg–Landau equations

In the paper, we study two-dimensional nonlinear spatial fractional complex Ginzburg–Landau equations. A centered finite difference method is exploited to discretize the spatial variables, while a three-level finite difference scheme is applied for the time integration. Theoretically, we prove the proposed method is uniquely solvable and unconditionally stable, with second order accuracy on both time and space, respectively. As the resulting discretized systems possess the block-Toeplitz structure, we proposed the preconditioned GMRES method with a block circulant preconditioner to speed up the convergence rate of the iteration. Meanwhile, fast Fourier transformation is utilized to reduce the complexity for calculating the discretized systems. Numerical experiments are carried out to verify the theoretical results and demonstrate that the proposed method enjoys the excellent computational advantage.

On the convergence of difference schemes for the generalized BBM–Burgers equation

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order k - 1 {k-1} when the exact solution belongs to the Sobolev space W 2 k ⁢ ( Q ) {W_{2}^{k}(Q)} , 1 < k ≤ 3 {1<k\leq 3} .

Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-Up Problem

AbstractWe consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, whose solution becomes unbounded in a finite time T. The finite time T is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in this paper a finite difference scheme with a rather larger grid size and show the convergence of the numerical solution and the numerical blow-up time. Application to the nonlinear wave equation is also discussed.

A Second-Order Three-Level Difference Scheme for a Magneto-Thermo-Elasticity Model

AbstractThis article deals with the numerical solution to the magneto-thermo-elasticity model, which is a system of the third order partial differential equations. By introducing a new function, the model is transformed into a system of the second order generalized hyperbolic equations. A priori estimate with the conservation for the problem is established. Then a three-level finite difference scheme is derived. The unique solvability, unconditional stability and second-order convergence inL∞-norm of the difference scheme are proved. One numerical example is presented to demonstrate the accuracy and efficiency of the proposed method.

Investigation of dual-phase-lag heat conduction model in a nanoscale metal-oxide-semiconductor field-effect transistor

This paper investigates the numerical simulation of non-Fourier transient heat transfer in a two-dimensional sub-100nm metal-oxide-semiconductor field-effect transistor (MOSFET). The dual-phase-lag (DPL) model with a specific normalization procedure is introduced for the modeling of nanoscale heat transport. The boundary conditions are selected similar to what existed in a real MOSFET device, both uniform and non-uniform heat generations within the transistor are applied, and the end parts of the top boundary which are in contact with the metallic material are left open. A temperature-jump boundary condition is used on all boundaries in order to consider the boundary phonon scattering at micro and nanoscale. A three-level finite difference scheme has been employed to generate the numerical results which are illustrated for a silicon sub-100nm MOSFET corresponding to the Knudsen number of 10. The results are presented at real times less than 50ps to avoid the three-dimensional effects. It is concluded that the combination of the DPL model with mixed-type temperature boundary condition is able to predict the heat flux and temperature distribution obtained from the Boltzmann transport equation (BTE) more accurate than the ballistic-diffusive equations (BDE). After verification of our results, the thermal field and the hotspot temperature within the bulk silicon are presented and the effect of adding a layer of silicon-dioxide to the transistor on its thermal behavior has been investigated.

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

Analytical and numerical solutions of the 2D transient dual-phase-lag (DPL) heat conduction equation are presented in this article. The geometry is that of a simplified metal oxide semiconductor field effect transistor with a heater placed on it. A temperature jump boundary condition is used on all boundaries in order to consider boundary phonon scattering at the micro- and nanoscale. A combination of a Laplace transformation technique and separation of variables is used to solve governing equations analytically, and a three-level finite difference scheme is employed to generate numerical results. The results are illustrated for three Knudsen numbers of 0.1, 1, and 10 at different instants of time. It is seen that the wave characteristic of the DPL model is strengthened by increasing the Knudsen number. It is found that the combination of the DPL model with the proposed mixed-type temperature boundary condition has the potential to accurately predict a 2D temperature distribution not only within the transistor itself but also in the near-boundary region.

Dynamic PDE parametric curves

Dynamic partial differential equation (PDE) parametric curves which can be expressed as a coupled system of two hyperbolic equations are developed. In curve design, dynamic PDE parametric curves can be modified intuitively and are more flexible than ordinary differential equation (ODE) curves. The calculation of dynamic PDE parametric curves must recur to numerical methods and a three-level finite difference scheme is proposed. Approximation and stability properties for the scheme are proved and convergence property is derived. An example of interpolating PDE curves is presented as an application of dynamic PDE parametric curves.

A finite difference scheme for solving parabolic two-step micro-heat transport equations in a double-layered micro-sphere heated by ultrashort-pulsed lasers

Ultrashort-pulsed lasers with pulse durations of the order of sub-picosecond to femtosecond domain possess exclusive capabilities in limiting the undesirable spread of the thermal process zone in the heated sample. Parabolic two-step micro heat transport equations have been widely applied for thermal analysis of thin metal films exposed to picosecond thermal pulses. In this study, we develop a three-level finite difference scheme for solving the micro heat transport equations in a double-layered micro sphere. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results for thermal analysis of a gold layer coated on a chromium padding layer are obtained. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

A stable three-level finite difference scheme for solving the parabolic two-step model in a 3D micro-sphere heated by ultrashort-pulsed lasers

Ultrashort-pulsed lasers with pulse durations of the order of sub-picosecond to femtosecond domain possess exclusive capabilities in limiting the undesirable spread of the thermal process zone in the heated sample. Parabolic two-step micro heat transport equations have been widely applied for thermal analysis of thin metal films exposed to picosecond thermal pulses. In this study, we develop a three level finite difference scheme for solving the heat transport equations in a three-dimensional micro-sphere heated by ultrashort-pulsed lasers. It is shown that the scheme is unconditionally stable. The method is illustrated by numerical examples.

A STABLE THREE-LEVEL FINITE-DIFFERENCE SCHEME FOR SOLVING A THREE-DIMENSIONAL DUAL-PHASE-LAGGING HEAT TRANSPORT EQUATION IN SPHERICAL COORDINATES

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation because a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in three-dimensional spherical coordinates and develop a three-level finite-difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is unconditionally stable. The method is illustrated by numerical examples.

A stable and convergent three-level finite difference scheme for solving a dual-phase-lagging heat transport equation in spherical coordinates

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is unconditionally stable and convergent. The method is illustrated by two numerical examples.

A convergent three‐level finite difference scheme for solving a dual‐phase‐lagging heat transport equation in spherical coordinates

AbstractHeat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second‐order derivative of temperature with respect to time and a third‐order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in spherical coordinates and develop a three‐level finite difference scheme for solving the heat transport equation in a microsphere. It is shown that the scheme is convergent, which implies that the scheme is unconditionally stable. Results show that the numerical solution converges to the exact solution. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 60–71, 2004.

A THREE-LEVEL FINITE-DIFFERENCE SCHEME FOR SOLVING MICRO HEAT TRANSPORT EQUATIONS WITH TEMPERATURE-DEPENDENT THERMAL PROPERTIES

Heat transport at the microscale is important for the processing of materials with a pulsed laser. In this study, we develop a three-level finite-difference scheme for solving micro heat transport equations with temperature-dependent thermal properties obtained based on the parabolic two-step model. It is shown by the discrete energy method that for constant thermal properties the scheme is unconditionally stable. Numerical results for thermal analysis of a gold film are obtained.

An implicit finite difference approximation to the one-dimensional transport equation

The properties of an implicit three-level finite difference scheme are investigated. The modified equivalent partial differential equation is used to determine the speed and amplitude characteristics of the scheme, and these are used to determine the optimal value of the weighting between a central difference and an upwind biased difference. Results of numerical experiments that confirm that the predicted value of the weight does minimise the error are presented.

The numerical solution of a singularly perturbed problem for semilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.