Numerical Solution Of Parabolic Equations Research Articles

Numerical Solution of Parabolic in Partial Differential Equations (PDEs) in One and Two Space Variable

In this paper, we shall be concerned with the numerical solution of parabolic equations in one space variable and the time variable t. We expand Taylor series to derive a higher-order approximation for Ut. We begin with the simplest model problem, for heat conduction in a uniform medium. For this model problem, an explicit difference method is very straightforward in use, and the analysis of its error is easily accomplished by the use of a maximum principle. As we shall show, however, the numerical solution becomes unstable unless the time step is severely restricted, so we shall go on to consider other, more elaborate, numerical methods which can avoid such a restriction. The additional complication in the numerical calculation is more than offset by the smaller number of time steps needed. We then extend the methods to problems with more general boundary conditions, then to more general linear parabolic equations. Finally, we shall discuss the more difficult problem of the solution of nonlinear equations.

A reduced fourth-order compact difference scheme based on a proper orthogonal decomposition technique for parabolic equations

In this paper, we present a reduced high-order compact finite difference scheme for numerical solution of the parabolic equations. CFDS4 is applied to attain high accuracy for numerical solution of parabolic equations, but its computational efficiency still needs to be improved. Our approach combines CFDS4 with proper orthogonal decomposition (POD) technique to improve the computational efficiency of the CFDS4. The validation of the proposed method is demonstrated by four test problems. The numerical solutions are compared with the exact solutions and the solutions obtained by the CFDS4. Compared with CFDS4, it is shown that our method has greatly improved the computational efficiency without a significant loss in accuracy for solving parabolic equations.

Tailored Finite Point Method for Parabolic Problems

Abstract In this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.

Improved accuracy for time-splitting methods for the numerical solution of parabolic equations

In this work, we study time-splitting strategies for the numerical approximation of evolutionary reaction–diffusion problems. In particular, we formulate a family of domain decomposition splitting methods that overcomes some typical limitations of classical alternating direction implicit (ADI) schemes. The splitting error associated with such methods is observed to be O(τ2) in the time step τ. In order to decrease the size of this splitting error to O(τ3), we add a correction term to the right-hand side of the original formulation. This procedure is based on the improved initialization technique proposed by Douglas and Kim in the framework of ADI methods. The resulting non-iterative schemes reduce the global system to a collection of uncoupled subdomain problems that can be solved in parallel. Computational results comparing the newly derived algorithms with the Crank–Nicolson scheme and certain ADI methods are presented.

Wavelet Method for Numerical Solution of Parabolic Equations

We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

Solving the heat equation on the unit sphere via Laplace transforms and radial basis functions

We propose a method to construct numerical solutions of parabolic equations on the unit sphere. The time discretization uses Laplace transforms and quadrature. The spatial approximation of the solution employs radial basis functions restricted to the sphere. The method allows us to construct high accuracy numerical solutions in parallel. We establish $L_2$ error estimates for smooth and nonsmooth initial data, and describe some numerical experiments.

Numerical Solution of Parabolic Equation on a 2-D Infinite Strip with Composite Hermite-Legendre Spectral Method

For solving numerically parabolic differential equation on a two-dimensional infinite strip, composite Hermite-Legendre Galerkin method is proposed in this article. By making use of stabilised scaled factor, the proposed method achieves stability. We also establish the convergence result for the proposed method. Numerical tests conduct for the model problem. It is shown that the proposed method is efficient.

A reduced-order LSMFE formulation based on POD method and implementation of algorithm for parabolic equations

Proper orthogonal decomposition (POD) technology has been successfully used in the reduced-order modeling of complex systems. In this paper, we extend the applications of POD method to a usual least-squares mixed finite element (LSMFE) formulation for two-dimensional parabolic equations with real practical applied background. POD bases here are constructed using the method of snapshots. Karhunen–Loeve projection is used to develop a reduced-order LSMFE formulation obtained by projecting the usual LSMFE formulation onto the most important POD bases. The reduced-order LSMFE formulation has lower dimensions and sufficiently high accuracy, is suitable for finding approximate solutions for two-dimensional parabolic equations, and can circumvent the constraint of the convergence stability what is called Brezzi–Babuška condition. We derive the error estimates between the reduced-order LSMFE solutions and the usual LSMFE solutions for parabolic equations and provide the implementation of algorithm for solving the reduced-order LSMFE formulation so as to supply scientific theoretic basis and computing way for practical applications. Two numerical examples are used to validate that the results of numerical computations are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order LSMFE formulation based on POD method is feasible and efficient for finding numerical solutions of parabolic equations.

Numerical solution of parabolic equations in high dimensions

We consider the numerical solution of diffusion problems in (0,T ) x Ω for and for T > 0 in dimension d d ≥ 1. We use a wavelet based sparse grid space discretization with mesh-width h and order p d ≥ 1, and hp discontinuous Galerkin time-discretization of order on a geometric sequence of many time steps. The linear systems in each time step are solved iteratively by GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2 (Ω)-error of O(N-p ) for u(x,T) where N is the total number of operations, provided that the initial data satisfies with e > 0 and that u(x,t) is smooth in x for t>0 . Numerical experiments in dimension d up to 25 confirm the theory.

Efficient Parallel Algorithms for Parabolic Problems

Domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation. Implicit schemes are used in order to march with large time steps. Parallelization is realized by approximating interface values using explicit computation. Various techniques are examined, including a multistep second order explicit scheme and a one-step high-order scheme. We show that the resulting schemes are of second order global accuracy in space, and stable in the sense of Osher or in $L_{\infty }$. They are optimized with respect to the parallel efficiency.

Stabilized Explicit-Implicit Domain Decomposition Methods for the Numerical Solution of Parabolic Equations

We report a class of stabilized explicit-implicit domain decomposition (SEIDD) methods for the numerical solution of parabolic equations. Explicit-implicit domain decomposition (EIDD) methods are globally noniterative, nonoverlapping domain decomposition methods, which, when compared with Schwarz-algorithm-based parabolic solvers, are computationally and communicationally efficient for each simulation time step but suffer from small time step size restrictions. By adding a stabilization step to EIDD, the SEIDD methods retain the time-stepwise efficiency in computation and communication of the EIDD methods but exhibit much better numerical stability. Three SEIDD algorithms are presented in this paper, which are experimentally tested to show excellent stability, computation and communication efficiencies, and high parallel speedup and scalability.

A numerical study of mixed parabolic–gradient systems

This paper is concerned with the numerical solution of parabolic equations coupled with gradient equations. The gradient equations are ordinary differential equations whose solutions define positions of particles in the spatial domain of the parabolic equations. The vector field of the gradient equations is determined by gradients of solutions to the parabolic equations. Such mixed parabolic–gradient systems are for example, used in neurobiological studies of the formation of axonal connections in the nervous system. We discuss a numerical approach for solving parabolic–gradient systems on a grid. The basic ingredients are the fourth-order spatial finite differencing for the parabolic equations, piecewise cubic Hermite interpolation for approximating the gradient equations, and explicit time-stepping by means of a Runge–Kutta–Chebyshev method.

Algebraic approximants and the numerical solution of parabolic equations

The heat equation is but one example of problems which involve multiple scales. There is a lot of transient behavior which for many problems is of no particular interest. What is of concern is the long time-scale behavior. However the presence of the short time-scale behavior would seem to require numerical integration methods to take very short time steps to follow the behavior accurately. For these problems, what is desired is a numerical method which is accurate for the long time-scale behavior, and causes the transients to die out quickly. That their rate of decay is not quite right is not important for this class of problems. A formalism is developed which allows the straightforward derivation of finite-difference schemes which involve several prior times from algebraic approximants. The algebraic approximants turn out to be, in a quite natural way, approximations to the function exp{−4μ[ arcsin( w/4 )] 2} where μ= κΔt/( Δx) 2 is the Courant number. Several of the simpler cases are investigated, and, of the implicit schemes, a couple are found to be, not only of higher order accuracy than most currently popular schemes, but also unconditionally stable and in fact unconditionally stiff stable! The higher order accuracy and stiff stability properties are just what is required for this sort of problem.

Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method

The numerical solution of a parabolic equation with memory is considered. The equation is first discretized in time by means of the discontinuous Galerkin method with piecewise constant or piecewise linear approximating functions. The analysis presented allows variable time steps which, as will be shown, can then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data. The combination with finite element discretization in space is also studied.

Influence of the misalignment of mirrors and of a deviation of the shapes of their reflecting surfaces from ideal on the losses and mode structure of the field in an unstable resonator with a ring-shaped active region and an internalWaxicon

The scalar theory of diffraction and a numerical solution of parabolic equations by the method of finite elements are used to find the eigenvalues representing the radiation losses in resonators containing a ring-shaped active region, an axicon, and mirrors with smooth edges. Calculations are reported also of the structures—corresponding to these losses—of the lowest modes (fundamental TEM00 and first azimuthal TEM10). These structures depend on the nature of the deformation of the reflecting surfaces of the mirrors and on the mirror misalignment. It is shown that the losses and the mode structure are strongly affected by the angle of tilt of a plane mirror located in the ring-shaped active region of the resonator and by the oval shape of the reflecting conical surfaces.

Physical control of numerical solution of parabolic equations

A simple, efficient, powerful numerical method for solving linear, one-dimensional parabolic differential equations is described. The method consists in placing external disturbances whose intensities are found by recurrence so as to meet specified boundary conditions by collocation. The equation can have variable coefficients and the boundaries can move as continuous functions of time. Boundary conditions must be piecewise continuous. The method does not always converge and when it does it may converge to a wrong answer. Sufficient conditions are given for convergence; they are also necessary for certain large classes of problems. Necessary and sufficient conditions are also given for convergence to be to the correct answer. When these are not met it is possible to adjust the answer to which the method converges so that this answer coincides with the exact values. One way is to monitor physical quantities. In so doing one can simultaneously correct for the effects of having erroneously estimated the parameters that characterise the medium. The method and manner of adjustment are illustrated by means of examples in soil consolidation. It is found that use of least squares is inadequate as a criterion of goodness of fit in problems governed by parabolic differential equations.

Physical control of numerical solution of parabolic equations

Physical control of numerical solution of parabolic equations

A splitting procedure for parabolic equations of higher order

In this paper, the numerical solution of parabolic equations of order n is shown to be easily accomplished by a splitting procedure involving the use of n computational nets.

On the Numerical Solution of Parabolic Equations in a Single Space Variable by the Continuous Time Galerkin Method

We consider the Galerkin method to solve a parabolic initial boundary value problem in one space variable, using piecewise polynomial functions and give an alternative proof of superconvergence. Then by means of Lobatto quadrature, we obtain purely explicit vector initial value problems without loss in the order of accuracy, global or pointwise.

Some computational aspects in the numerical solution of parabolic equations

Experimental studies of different questions related to the numerical solution of parabolic equations have been untertaken: 1. (i) the choice of the value of θ in the θ-method is discussed, and 2. (ii) the number of iterations in the solution of algebraic equations with matrix of the form M + (1 − θ) τ K with M mass matrix, K stiffness matrix, has been studied, both for linear and for nonlinear problems.