Three-level Finite Difference 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.

A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh\u2013Stokes problem for a heated generalized second-grade fluid

In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

On the identification of nonlinear terms in the generalized Camassa-Holm equation involving dual-power law nonlinearities

The nonlinear waves constitute a significant concern in the design in order to understand wave behaviors and structures. Due to the complexity of the waves phenomenon, many mathematical models have been coming up and extensively being studied until now. In this paper, we introduce a new aspect to study a wave model by coupling of the classical Camassa-Holm equation and the BBM-KdV equation with the dual-power law nonlinearities. The new model still satisfies the fundamental energy conservative property as the original models. As we have known, the use of an efficient and accurate numerical method can be a beneficial tool for studying the wave behaviors; therefore, we aim to propose a finite difference scheme for solving the new model. A three-level finite difference technique is applied here in order to be able to design a linear scheme. The highlight of our scheme is to propose an alternative way to discretize the highly nonlinear terms. The class of analytical solutions of the model is derived and used to validate our numerical scheme. Like the design, the energy conservation law is preserved through the numerical scheme. In the context of theoretical analysis, the existence and uniqueness is analyzed based on a priori estimate. The stability and convergence of the numerical solution with second-order accuracy on both space and time are also provided under a mild restriction on the ratio of τ/h. Many comparisons of our scheme and a modification of the existing finite difference scheme are preformed, and the consequences confirm that the proposed scheme gives a significant improvement over the other. Moreover, in the numerical simulations, the faithfulness of the proposed method is validated by the pieces of evidence of depression solitary waves under the effect of the power of nonlinearities.

High order numerical schemes for solving fractional powers of elliptic operators

In many recent applications when new materials and technologies are developed it is important to describe and simulate new nonlinear and nonlocal diffusion transport processes. A general class of such models deals with nonlocal fractional power elliptic operators. In order to solve these problems numerically it is proposed (Vabishchevich, 2015) to consider equivalent local nonstationary initial value pseudo-parabolic problems. Previously such problems were solved by using the standard implicit backward and symmetrical Euler methods. In this paper we use the one-parameter family of three-level finite difference schemes for solving the initial value problem for the first order nonstationary pseudo-parabolic problem. The fourth-order approximation scheme is developed by selecting the optimal value of the weight parameter. The results of the theoretical analysis are supplemented by results of extensive computational experiments.

Novel algorithm based on modification of Galerkin finite element method to general Rosenau-RLW equation in (2 + 1)-dimensions

In this research, the presented method combines advantages of the finite element and three-level finite difference techniques to obtain the solution of the two-dimensional general Rosenau-RLW equation. An important point is that nevertheless the two-dimensional general Rosenau-RLW is a non-linear equation, but with the proposed scheme we are able to linearize this system and efficiently solve it due to the implicit nature of the system of equations. In addition, the existence and uniqueness of the approximate solution are approved. The convergence and stability of the approximate solution are also examined. The general formulation of the procedure is given and numerical results are carried out to confirm the accuracy of our theoretical results and the efficiency of the scheme.

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.

Non-Fourier effect of laser-mediated thermal behaviors in bio-tissues: A numerical study by the dual-phase-lag model

Pulsed laser has been widely used for the thermotherapy of skin diseases. In this process, non-Fourier heat conduction has to be considered because of its small time scale. After optical transmission and energy deposition were obtained by Monte Carlo method, the dual-phase-lag (DPL) model was employed and solved by using a three-level finite difference method to characterize non-Fourier heat conduction by focusing on the thermal lag during the interaction of pulsed laser with biological tissues. The following main conclusions are obtained. Substantial discrepancies exist among the temperature responses of biological tissues with and without considering the non-Fourier effect. In the DPL prediction, nonlinear temperature increases as laser pulse heating is performed, and the appearance of the highest temperature exhibits lagging behaviors. This is mainly dominated by the relaxation parameter of heat flux vector (τq). The lag later appears about 2.8 times the pulses when τq is equal to the pulse duration. τq and τT, which are respectively the relaxation parameters of heat flux vector and temperature gradient, can decrease the calculated temperature, whereas τT can equalize the temperature gradient at tissue interface. In the DPL model, the thermal relaxation times of the material do not stand for τq and τT, which are inherently related to the physical time scale.

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.

Two-level finite difference scheme of improved accuracy order for time-dependent problems of mathematical physics

In the theory of finite difference schemes, the most complete results concerning the accuracy of approximate solutions are obtained for two- and three-level finite difference schemes that converge with the first and second order with respect to time. When the Cauchy problem is numerically solved for a system of ordinary differential equations, higher order methods are often used. Using a model problem for a parabolic equation as an example, general requirements for the selection of the finite difference approximation with respect to time are discussed. In addition to the unconditional stability requirements, extra performance criteria for finite difference schemes are presented and the concept of SM stability is introduced. Issues concerning the computational implementation of schemes having higher approximation orders are discussed. From the general point of view, various classes of finite difference schemes for time-dependent problems of mathematical physics are analyzed.

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 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.

Three-level techniques for one-dimensional parabolic equation with nonlinear initial condition

In this paper a boundary value problem for one-dimensional heat equation is considered under the constraint of a nonlocal initial condition. In place of the classical specification of initial data, we impose the nonlocal initial condition. Efficient higher-order algorithms are developed for solving this parabolic partial differential equation with nonstandard initial condition. Several three-level finite difference schemes are presented. These schemes are based on the modification of the centred-time centred-space explicit formula, the three-level (1,3,1) explicit technique, the sixth-order (1,3,3) explicit method and the Dufort–Frankel finite difference scheme. Numerical computations combined with a simple iteration procedure of some examples are given to demonstrate the efficiency and accuracy of the new algorithms.

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.