Two-dimensional Parabolic Partial Differential Equations Research Articles

Computer simulation of the pulse-periodic electric field effect on the 2D director orientation of nematic liquid crystal. Experimental research of multimode nematic liquid crystal waveguides.

In this paper, we numerically investigate a two-dimensional differential equation describing the motion of a director of a nematic liquid crystal for the case of an alternating external electric field. The presence of the previously discovered accumulation effect has been confirmed by numerical modeling. A comparison is made with the case of a constant electric field, and also a qualitative comparison with an experiment is given. Incomplete agreement with experimental data indicates the need for further research. However, it should be noted that the constructed mathematical model of the phenomenon allows at this stage to obtain estimates that are sufficiently acceptable for experiment and correctly predict the dynamics of processes in liquid crystals. An analysis of the features of the propagation of quasi-waveguide modes in a liquid crystal waveguide showed that, in the case of dynamic processes, such effects as power exchange between coupled modes, leakage of modes, re-emission of modes into modes of a different order, etc., can be observed. The programs for numerical solution and computer modeling of two-dimensional parabolic partial differential equation were developed both in FORTRAN and C/C++. The results obtained are important for further investigation of dynamic processes inside non-stationary liquid crystal layers, both from a theoretical point of view for understanding kinetic processes in liquid crystals and from a practical point of view when organizing and conducting different experimental research.

A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite difference scheme whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases, however, it can easily be extended to other types of nonlinearities as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method.

Numerical Modeling of the Static Electric Field Effect on the Director of the Nematic Liquid Crystal Director

A two-dimensional model of the Frederiks effect is used to investigate the static electric field effect on the orientation of the nematic liquid crystal (LC) director in a side-electrode cell. The solutions are obtained by the standard finite-difference methods. The programs for the numerical solution of a two-dimensional parabolic partial differential equation are developed both in FORTRAN and C/C++. The Frederiks transition threshold for the central part of the cell and the dependences of the director’s orientation distribution on a high electric field are obtained. The results of the calculation are compared with the experimental data.

High order numerical algorithms based on biquadratic spline collocation for two-dimensional parabolic partial differential equations

ABSTRACTWe report a new algorithm for solving linear parabolic partial differential equations in two space dimension. The algorithm employs optimal biquadratic spline collocation for space discretization and modified trapezoidal rule for time discretization. We need to solve a block tridiagonal linear system at each time step, and obtain an approximate solution with error at space-time grid points. We analyse the stability of the new algorithm, and present a stability enhanced variant. Moreover, we give an acceleration strategy based on spectral deferred correction, and the theoretical accuracy can be increased to , where k is the number of correction loops. We also analyse the stability for the accelerated algorithms. Numerical experiments are attached to demonstrate the effectiveness of the new algorithms.

Numerical solution of Burgers’ equation by B-spline collocation

In this paper, the Burgers’ equation which is two-dimensional in space, time dependent parabolic differential equation was solved by b-spline collocation algorithms for solving two-dimensional parabolic partial differential equation. At first b-spline interpolation is introduced moreover, the numerical solution is represented as a bi-variate piecewise polynomial with unknown time-dependent coefficients are determined by requiring the numerical solution to satisfy the PDE at a number of points within the spatial domain i.e. we collocate simultaneously in both spatial dimensions. The accuracy of the proposed method is demonstrates by some test problems. The numerical results are found good agreement with exact solution.

Optimal control of two-dimensional parabolic partial differential equations with application to steel billets cooling in continuous casting secondary cooling zone

Optimal control of two-dimensional parabolic partial differential equations with application to steel billets cooling in continuous casting secondary cooling zone

An accurate Chebyshev pseudospectral scheme for multi-dimensional parabolic problems with time delays

In this paper, the Chebyshev Gauss-Lobatto pseudospectral scheme is investigated in spatial directions for solving one-dimensional, coupled, and two-dimensional parabolic partial differential equations with time delays. For the one-dimensional problem, the spatial integration is discretized by the Chebyshev pseudospectral scheme with Gauss-Lobatto quadrature nodes to provide a delay system of ordinary differential equations. The time integration of the reduced system in temporal direction is implemented by the continuous Runge-Kutta scheme. In addition, the present algorithm is extended to solve the coupled time delay parabolic equations. We also develop an efficient numerical algorithm based on the Chebyshev pseudospectral algorithm to obtain the two spatial variables in solving the two-dimensional time delay parabolic equations. This algorithm possesses spectral accuracy in the spatial directions. The obtained numerical results show the effectiveness and highly accuracy of the present algorithms for solving one-dimensional and two-dimensional partial differential equations.

B-Spline Gaussian Collocation Software for Two-Dimensional Parabolic PDEs

AbstractIn this paper we describe newB-spline Gaussian collocation software for solving two-dimensional parabolic partial differential equations (PDEs) defined over a rectangular region. The numerical solution is represented as a bi-variate piecewise polynomial (using a tensor productB-spline basis) with time-dependent unknown coefficients. These coefficients are determined by imposing collocation conditions: the numerical solution is required to satisfy the PDE and boundary conditions at images of the Gauss points mapped onto certain subregions of the spatial domain. This leads to a large system of time-dependent differential algebraic equations (DAEs) which is solved using the DAE solver, DASPK. We provide numerical results in which we use the new software, called BACOL2D, to solve three test problems.

A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets

In this paper we present two new numerically stable methods based on Haar and Legendre wavelets for one- and two-dimensional parabolic partial differential equations (PPDEs). This work is the extension of the earlier work [1–3] from one- and two-dimensional boundary-value problems to one- and two- dimensional PPDEs. Two generic numerical algorithms are derived in two phases. In the first stage a numerical algorithm is derived by using Haar wavelets and then in the second stage Haar wavelets are replaced by Legendre wavelets in quest for better accuracy. In the proposed methods the time derivative is approximated by first order forward difference operator and space derivatives are approximated using Haar (Legendre) wavelets. Improved accuracy is obtained in the form of wavelets decomposition. The solution in this process is first obtained on a coarse grid and then refined towards higher accuracy in the high resolution space. Accuracy wise performance of the Legendre wavelets collocation method (LWCM) is better than the Haar wavelets collocation method (HWCM) for problems having smooth initial data or having no shock phenomena in the solution space. If sharp transitions exists in the solution space or if there is a discontinuity between initial and boundary conditions, LWCM loses its accuracy in such cases, whereas HWCM produces a stable solution in such cases as well. Contrary to the existing methods, the accuracy of both HWCM and LWCM do not degrade in case of Neumann’s boundary conditions. A distinctive feature of the proposed methods is its simple applicability for a variety of boundary conditions. Performances of both HWCM and LWCM are compared with the most recent methods reported in the literature. Numerical tests affirm better accuracy of the proposed methods for a range of benchmark problems.

COMPONENTWISE SPLITTING METHODS FOR PRICING AMERICAN OPTIONS UNDER STOCHASTIC VOLATILITY

Efficient numerical methods for pricing American options using Heston's stochastic volatility model are proposed. Based on this model the price of a European option can be obtained by solving a two-dimensional parabolic partial differential equation. For an American option the early exercise possibility leads to a lower bound for the price of the option. This price can be computed by solving a linear complementarity problem. The idea of operator splitting methods is to divide each time step into fractional time steps with simpler operators. This paper proposes componentwise splitting methods for solving the linear complementarity problem. The basic componentwise splitting decomposes the discretized problem into three linear complementarity problems with tridiagonal matrices. These problems can be efficiently solved using the Brennan and Schwartz algorithm, which was originally introduced for American options under the Black and Scholes model. The accuracy of the componentwise splitting method is increased by applying the Strang symmetrization. The good accuracy and the computational efficiency of the proposed symmetrized splitting method are demonstrated by numerical experiments.

Application of the Adomian decomposition method for two-dimensional parabolic equation subject to nonstandard boundary specifications

In this work we are concerned with the solution of a nonlocal boundary value problem. An approach is presented for solving the two-dimensional parabolic partial differential equation subject to integral boundary specifications. The main objective is to propose an alternative method of solution, one not based on finite difference methods or finite element schemes or spectral techniques. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the two-dimensional linear parabolic partial differential equation with nonlocal boundary specifications replacing the classical boundary conditions. The Adomian decomposition method is used by many researchers to investigate several scientific applications and requires less work if compare with the traditional techniques. The introduction of this idea as will be discussed, not only provides the solution in a series form but it also guarantees considerable saving of the calculations volume. The solutions will be handle more easily, quickly and elegantly without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent power series with easily computed components. The Adomian decomposition scheme is easy to program in applied problems and provides immediate and convergent solutions without any need for linearization or discretization. To give a clear overview of the methodology, we have selected illustrative example.

An extended Krylov subspace method to simulate single-phase fluid flow phenomena in axisymmetric and anisotropic porous media

We develop and validate a novel numerical algorithm for the simulation of axisymmetric single-phase fluid flow phenomena in porous and permeable media. In this new algorithm, the two-dimensional parabolic partial differential equation for fluid flow is transformed into an explicit finite-difference operator problem. The latter is solved by making use of an extended Krylov subspace method (EKSM) constructed with both positive and inverse powers of the finite-difference operator. A significant advantage of the method of solution presented in this paper is that simulations of pressure can be obtained at a multitude of times with practically the same efficiency as that of a single-time simulation. Moreover, the usage of inverse powers of the finite-difference operator provides a substantial increase in efficiency with respect to that of standard Krylov subspace methods. Tests of numerical performance with respect to analytical solutions for point and line sources validate the accuracy of the developed method of solution. We also validate the algorithm by making comparisons between analytical and numerical solutions in the Laplace transform domain. Additional tests of accuracy and efficiency are performed against a commercial simulator for spatially complex and anisotropic models of permeable media.

Numerical procedures for a boundary value problem with a non-linear boundary condition

Numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial-boundary value problem for the two-dimensional parabolic partial differential equation (PDE) with a non-linear boundary condition. This class of parabolic PDEs plays a very important role in many branches of science and engineering. The non-linear condition is in the form of a double integral giving the specification of mass in the solution domain. Not only the problem has both Neumann and Dirichlet boundary conditions but the Dirichlet boundary condition is in a non-standard form. While sharing some common features with the one-dimensional models, the solution of two-dimensional equations are substantially more difficult, thus some considerations are taken to be able to extend some ideas of one-dimensional case. Due to the structure of the boundary conditions a reduced one-dimensional parabolic equation for the new unknown v(y,t)=∫10u(x,y,t)dx can be formulated. The resulting problem has a non-local boundary condition. The new two-dimensional parabolic PDE with Neumann’s boundary conditions will be solved numerically by using the method of lines semi-discretization approach. The space derivatives in the PDE are approximated by finite difference replacements. The solution of the resulting system of first-order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time is controlled by choosing several subdiagonal Pade approximants to replace this matrix exponential term. Numerical techniques are developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The article concludes with the results of some numerical experiments.

Analysis of cage induction motor by means of the finite element method and coupled system of field, circuit and motion equations

A mathematical model is presented for transient and steady-state electromechanical analysis of cage induction motor by using the finite element method. The model is based on the coupling of field, field-circuit, circuit and motion equations. The electromagnetic processes in the active zone of the motor are described by a field equation, which is two-dimensional parabolic partial differential equation, including the media nonlinearities, eddy currents and rotor motion. The influence of the stator and rotor end regions is taken into account by circuit elements.

A static-regridding method for two-dimensional parabolic partial differential equations

The subject of this paper belongs to the field of numerical solution of time-dependent partial differential equations. Attention is focussed on parabolic problems in two space dimensions the solutions of which possess sharp moving transitions in space and time, such as steep moving fronts and emerging and disappearing layers. For such problems, a space grid held fixed throughout the entire calculation can be computationally inefficient, since, to afford an accurate approximation, such a grid would easily have to contain a very large number of nodes. We consider a so-called static-regridding method that adapts the space grid using nested, locally uniform grids. The notion of locally uniform grid refinement is an example of “domain decomposition”, the general idea of which is to decompose the original physical or computational domain into smaller subdomains and to solve the original problem on these subdomains. Hence, our computational subdomains are nested, locally uniform space grids with nonphysical boundaries which are generated up to a level of refinement good enough to resolve the anticipated fine scale structures. This way a fine grid covering the entire physical domain can be avoided. We discuss several aspects occuring in a static-regridding algorithm like ours, such as the data structure, the regridding strategy and the determination of initial and boundary conditions at coarse-fine grid interfaces. We also present two numerical examples to demonstrate the performance of the method. The first example is hypothetical and is used to illustrate the convergence behaviour of the method. The second example originates from practice and describes a model combustion process.

A mathematical model for the three-dimentional ocean sound propagation

A model is developed mathematically to represent sound propagation in a three-dimensional ocean. The complete development is based on characteristics of the physical environment, mathematical theory, and computational accuracy. While the two-dimentional underwater acoustic wave propagation problem is not yet solved completely for range-dependent environments,three-dimentional environmental effects, such as fronts and eddies, often cannot be neglected. To predict underwater sound propagation, one usually deals with the solution of the Helmholtz (reduced wave) equation. This elliptical equation, along with a set of boundary conditions including a wall condition at the maximum range, forms a well-posed problem, which is pure boundary-value problem. An existing approach to economically solve this three-dimensional range-dependent problem is by means of a two-dimensional parabolic partial differential equation. This parabolic approximation approach, within the limitation of mathematical and acoustical approximations, offers efficient solutions to a class of long-range propagation problems. The parabolic wave equation is much easier to solve than the elliptic equation; one major saving is the removal of the wall boundary condition at the maximum range. The application of the two-dimensional parabolic wave equation to a number of realistic problems has been successful. We discuss the extension of the parabolic equation approach to three-dimensional problems. This paper begins with general considerations of the three-dimensional elliptic wave equation and shows how to transform this equation into parabolic equations which are easier to solve. The development of this paper focuses on wide angle three-dimensional underwater acoustic propagation and accommodates as a special case prevoius developments by other authors. In the course of our development, the physical properties, mathematical validity, and computational accuracy are the primary factors considered. We describe how parabolic wave equations are derived and how wide angle propagation is taken into consideration. Then, a discussion of the limitations and the advantages of the parabolic equation approximation is highlighted. These provide the background for the mathematical formulation of three-dimensional underwater acoustic wave propagation models. Modelling the mathematical solution to three-dimensional underwater acoustic wave propagation involves difficulties both in describing the theoretical acoustics and in performing the large scale computations. We have used the mathematical and physical properties of the problem to simplify considerably. Simplications allow us to introduce a three-dimensional mathematical model for underwater acoustic propagation predictions. Our wide angle three-dimensional parabolic equation model is theoretically justifiable and computationally accurate. This model offers a variety of capabilities to handle a class of long-range propagation problems under acoustical environments with three-dimensional variations.

Solving parabolic PDE's by a decomposition method

The paper deals with the application of the hybrid method of decomposition for the solution of two-dimensional parabolic partial differential equations with constant coefficients. The convergence and accuracy of the CSDT method are analysed, deriving the form of the relationship between ‘eigenvalues’ and the time step. Theoretical problems of this method of decomposition are mentioned. Also, the expression for computing the coefficients for various types of boundary problems has been derived. Practical applications of the method are presented. The cubic spline function interpolation is used for the continuous signal reproduction in the hybrid system. Some results obtained are presented at the end.