Predictor-corrector methods for parabolic partial differential equations

In this paper we extend predictor-corrector methods, commonly used for the numerical solution of ordinary differential equations (o.d.e.s), to parabolic partial differential equations (p.d.e.s), typically of the form u_t = au_xx + ?(u, u_x, x, t). We describe linear multistep methods for p.d.e.s, the nonlinear algebraic equations arising from implicit formulae being solved using a corrector analogous to those used for o.d.e.s. A sufficient condition for convergence of the iteration is then derived and is found, in most cases, to be far less restrictive than that obtained from the usual method of lines approach. Numerical results are presented to investigate the necessity of this condition. They also indicate that we can accelerate convergence by reducing the time increment. This allows us to achieve convergence within a prescribed number of iterations and so to construct PC^m methods corresponding to P(EC)^m methods for o.d.e.s. Numerical results are also given to test the absolute stability of the Crank-Nicolson corrector for various predictors P, and iterations, m.     

On comparing the accuracy of some finite difference methods for parabolic partial differential equations

Methods for comparing the accuracy of numerical methods for the solution of parabolic partial differential equations are briefly surveyed. Criteria for a more systematic comparison are proposed and applied to an accuracy comparison of the classical and the modified Saul'yev methods and the Crank–Nicholson method on a simple linear problem. The results obtained, which include observation of the existence of optimal spatial mesh size for the two Saul'yev methods, are discussed.     

Accurate finite difference approximations for the solution of parabolic partial differential equations by semi-discretization

The advantages in formulation and use of semi-discretized approaches to the numerical solution of initial/boundary value problems are well known. The aim of this paper is to demonstrate that it is feasible to obtain accurate results even with a coarse spatial mesh. A method is developed which produces in a simple manner matrix representations for high-order central difference operators. Dirichlet, Neumann and mixed boundary conditions are considered, both homogeneous and non-homogeneous. It is shown in all cases that, for linear problems at least, there is no need to use a finer mesh than that dictated by the essential frequency content of the initial function data.     

Parameter determination in parabolic partial differential equations from overspecified boundary data

Several different problems of determining unknown parameters or functions in parabolic partial differential equations from overspecified boundary data are treated. Explicit solutions are presented for various problems and the existence of a solution of various other problems is presented. The unknown parameters range over unknown sources, coefficients of $\text{u}$ and $\text{u}{}_{\mathrm{x}}$, terms in the equation, and coefficients in a boundary condition.     

具连续分布滞量的非线性抛物型偏微分方程的振动准则

考虑一类具连续分布滞量的非线性抛物型偏微分方程的振动性，借助Green定理将多维振动问题转化为关于某一类具连续分布滞量的非线性微分不等式的一维问题，给出了该类方程在Robin，Dirichlet边值条件下所有解振动的若干充分判据。所得结论充分地表明，振动是由时滞量引起的。     

On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations

Splitting methods for time-dependent partial differential equations usually exhibit a drop in accuracy if boundary conditions become time-dependent. This phenomenon is investigated for a class of splitting methods for two-space dimensional parabolic partial differential equations. A boundary-value correction discussed in a paper by Fairweather and Mitchell for the Laplace equation with Dirichlet conditions, is generalized for a wide class of initial boundary-value problems. A numerical comparison is made for the ADI method of Peaceman-Rachford and the LOD method of Yanenko applied to problems with Dirichlet boundary conditions and non-Dirichlet boundary conditions.     

On the construction of accurate difference schemes for hyperbolic partial differential equations

Summary Methods are developed for increasing the fidelity of difference approximations to hyperbolic partial differential equations. A relation between the truncation error and the exact and approximate amplification factors is derived. Based upon this relation, quantitative criteria for the minimization of dissipation and dispersion are derived, and difference schemes which satisfy these criteria are constructed. Completely new schemes, one of them promising, are obtained, together with several well-known schemes. One of these is the Fromm scheme, for which previously only a heuristic derivation could be given. It is shown that in general the accuracy of the Rusanov-Burstein-Mirin scheme is disappointing. A simple modification was found to remedy this deficiency.     

Comparative study of six explicit and two implicit finite difference schemes for solving one-dimensional parabolic partial differential equations

Eight finite difference schemes used in solving parabolic partial differential equations are compared with respect to accuracy, execution time and programming effort. The analysis presented is useful in selecting the appropriate numerical scheme depending on the emphasis placed upon accuracy, execution time or programming effort.     

Traveling wave solutions of nonlinear partial differential equations

Elementary transformations are utilized to obtain traveling wave solutions of some diffusion and wave equations, including long wave equations and wave equations the nonlinearity of which consists of a linear combination of periodic functions, either trigonometric or elliptic. In particular, a theorem is established relating the solutions of a single cosine equation and a double sine-cosine equation. It is shown that the latter admits a Bäcklund Transformation.     

On the existence and formulation of variational principles for systems of scalar partial differential equations

Using the consistency conditions due to Tonti for a potential operator, alternate variational principles for systems of scalar differential equations corresponding to the following physical situations have been formulated: (i) Isoenergetic flows, (ii) Plasma flows governed by modified Korteweg-de Vries type equation, (iii) Longitudinal waves in a non-linear geometrically dispersive fluid, (iv) Non-linear erosion.     

Domain mappings for the numerical solution of partial differential equations

Independent variable transformations of partial differential equations are examined with regard to their use in numerical solutions. Systems of first order and second order partial differential equations in conservative and nonconservative form are considered. These general equations are transformed using generalized mapping functions and important computational features of the transformed equations are discussed. Examples of mappings which regularize domains are given involving various types of partial differential equations. These mappings are of particular importance in finite difference approximations because of the ease with which a mesh can be adapted to regions formed by co-ordinate lines.     

An introduction to partial differential equations constrained optimization

Partial differential equation (PDE) constrained optimization is designed to solve control, design, and inverse problems with underlying physics. A distinguishing challenge of this technique is the handling of large numbers of optimization variables in combination with the complexities of discretized PDEs. Over the last several decades, advances in algorithms, numerical simulation, software design, and computer architectures have allowed for the maturation of PDE constrained optimization (PDECO) technologies with subsequent solutions to complicated control, design, and inverse problems. This special journal edition, entitled “PDE-Constrained Optimization”, features eight papers that demonstrate new formulations, solution strategies, and innovative algorithms for a range of applications. In particular, these contributions demonstrate the impactfulness on our engineering and science communities. This paper offers brief remarks to provide some perspective and background for PDECO, in addition to summaries of the eight papers.     

Solution of nonlinear partial differential equations using the Chebyshev spectral method

The spectral method is a powerful numerical technique for solving engineering differential equations. The method is a specialization of the method of weighted residuals. Trial functions that are easily and exactly differentiable are used. Often the functions used also satisfy an orthogonality equation, which can improve the efficiency of the approximation. Generally, the entire domain is modeled, but multiple sub-domains may be used. A Chebyshev-Collocation Spectral method is used to solve a two-dimensional, highly nonlinear, two parameter Bratu's equation. This equation previously assumed to have only symmetric solutions are shown to have regions where solutions that are non-symmetric in x and y are valid. Away from these regions an accurate and efficient technique for tracking the equation's multi-valued solutions was developed.It is found that the accuracy of the present method is very good, with a significant improvement in computer time.     

Point interpolation collocation method for the solution of partial differential equations

This paper presents a truly meshless method for solving partial differential equations based on point interpolation collocation method (PICM). This method is different from the previous Galerkin-based point interpolation method (PIM) investigated in the papers [G.R. Liu, (2002), mesh free methods, Moving beyond the Finite Element Method, CRC Press. G.R. Liu, Y.T. Gu, A point interpolation method for two-dimension solids, Int J Numer Methods Eng, 50, 937–951, 2001. G.R. Liu, Y.T. Gu, A matrix triangularization algorithm for point interpolation method, in Proceedings Asia-Pacific Vibration Conference, Bangchun Weng Ed., November, Hangzhou, People's Republic of China, 2001a, 1151–1154. 1–3.], because it is based on collocation scheme. In the paper, polynomial basis functions have been used. In addition, Hermite-type interpolations called as inconsistent PIM has been adopted to solve PDEs with Neumann boundary conditions so that the accuracy of the solution can be improved. Several examples were numerically analysed. These examples were applied to solve 1D and 2D partial differential equations including linear and non-linear in order to test the accuracy and efficiency of the presented method based on polynomial basis functions. The h-convergence rates were computed for the PICM based on different model of regular and irregular nodes. The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.     

A numerical treatment to the solution of quasiparabolic partial differential equations

This paper presents results obtained by the implementation of a hybrid Laplace transform finite element method to the solution of quasiparabolic problem. The present method removes the time derivatives from the quasiparabolic partial differential equation using the Laplace transform and then solves the associated equation with the finite element method. The numerical inverse of the Laplace transform is realized by solving linear overdetermined systems and a polynomial equation of the kth order. Test examples are used to show that the numerical solution is comparable to the exact solution of the initial-boundary value problem at the given grid points.     

Three-dimensional eddy current problems using the T-Ω method and fredholm integral equations

A technique is discussed for computing three-dimensional magnetic fields created by an oscillating source in a conducting medium. The technique's contribution lies in the manner in which the Fredholm integral technique is merged with a representation of the magnetic field as the sum of a vector and the gradient of a scalar (the T-Ω representation). The T-Ω representation is shown to conveniently realize the boundary conditions without introducing higher order derivative terms into the integral equations. The technique offers a powerful means of numerically calculating three-dimensional fields, necessitating only the discretization of interfacial surfaces.     

On meromorphic complex differential equations

In this paper, we study first-order, autonomous, complex differential equations of the form i = f (z), where f (z) is the meromorphic function of the complex variable z, defined in a simply connected domain on the Riemann sphere. We concentrate on the phase portraits of such systems, with particular attention being paid to the existence and properties of closed orbits, and orbits which reach the point at infinity (or blow-up) in finite time. Applications of the general theoy are given, as well as discussion of higher dimensional systems     

A qualitative analysis of the behaviour of the Galerkin equations relevant to non-linear eddy current problems

The Galerkin equations relevant to the eddy currents induced in an iron body are considered. These equations are obtained by formulating the field problem in terms of the magnetic vector potential and by applying the Galerkin method. They are shown to have a unique steady-state solution if a certain condition on the magnetic constituitive relationship is satisfied. In particular a T-periodic source gives rise to a unique T-periodic solution to which all other solutions converge asymptotically independently from the initial conditions. Under the same condition the exponential decay of the ‘transients’ is shown, and an explicit lower and upper bound for its rate is given. These structural properties allow us to exclude a priori that qualitatively different asymptotic behaviours, including even chaotic solutions, may occur. Numerical simulation, when based on qualitative information of this type, enables us to obtain the quantitative properties in an efficient manner. In order to demonstrate the practical use of these results some numerical experiments are presented.