The numerical solution of linear time-dependent partial differential equations by the Laplace transform and finite difference approximations

A feasible method is presented for the numerical solution of a large class of linear partial differential equations which may have source terms and boundary conditions which are time-varying. The Laplace transform is used to eliminate the time-dependency and to produce a subsidiary equation which is then solved in complex arithmetic by finite difference methods. An effective numerical Laplace transform inversion algorithm gives the final solution at each spatial mesh point for any specified set of values of t. The single-step property of the method obviates the need to evaluate the solution at a large number of unwanted intermediate time points. The method has been successfully applied to a variety of test problems and, with two alternative numerical Laplace transform inversion algorithms, has been found to give results of good to excellent accuracy. It is as accurate as other established finite difference methods using the same spatial grid. The algorithm is easily programmed and the same program handles equations of parabolic and hyperbolic type.     

The numerical solution of non-linear partial differential equations by the method of lines

A new method for solving non-linear parabolic partial differential equations, based on the method of lines, is developed. Second order and fourth order finite difference approximations to the spatial derivatives are used, and in each case the band structure of the associated Jacobian is exploited. This, together with the use of Gear's fourth order stiffly stable method in Nordsieck form, leads to a method which compares favourably with the respected Sincovec and Madsen method on Burgers' equation. The method has been tested on a number of difficult problems in the literature and has proved to be most successful.     

Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations

The preconditioned Barzilai-Borwein method is derived and applied to the numerical solution of large, sparse, symmetric and positive definite linear systems that arise in the discretization of partial differential equations. A set of well-known preconditioning techniques are combined with this new method to take advantage of the special features of the Barzilai-Borwein method. Numerical results on some elliptic test problems are presented. These results indicate that the preconditioned Barzilai-Borwein method is competitive and sometimes preferable to the preconditioned conjugate gradient method.This author was partially supported by the Parallel and Distributed Computing Center at UCV.This author was partially supported by BID-CONICIT, project M-51940.     

Projection methods for the numerical solution of non-self-adjoint elliptic partial differential equations

We compare the relative performances of two iterative schemes based on projection techniques for the solution of large sparse nonsymmetric systems of linear equations, encountered in the numerical solution of partial differential equations. The Block–Symmetric Successive Over-Relaxation (Block-SSOR) method and the Symmetric–Kaczmarz method are derived from the simplest of projection methods, that is, the Kaczmarz method. These methods are then accelerated using the conjugate gradient method, in order to improve their convergence. We study their behavior on various test problems and comment on the conditions under which one method would be better than the other. We show that while the conjugate-gradient-accelerated Block-SSOR method is more amenable to implementation on vector and parallel computers, the conjugate-gradient accelerated Symmetric–Kaczmarz method provides a viable alternative for use on a scalar machine.     

A survey of dynamically-adaptive grids in the numerical solution of partial differential equations

The construction of dynamically-adaptive curvilinear coordinate systems based on numerical grid generation and the use thereof in the numerical solution of partial differential equations is surveyed, and correlations are made among the various approaches. These adaptive grids are coupled with the physical solution being done on the grid so that the grid points continually move in the course of the solution in order to resolve developing gradients, or higher variations, in the solution. Particular attention is given to systems using elliptic grid generation based on variational principles. It is noted that dynamic grid adaption can remove the oscillations common when strong gradients occur on fixed grids, and that it appears that when the grid adapts to the solution most numerical solution algorithms work well. Particular applications in computational fluid dynamics and heat transfer are noted.     

The numerical solution of unstable ordinary differential equations

Invariant imbedding, or the Riccati transformation, has been used to solve unstable ordinary differential equations for a few years. This paper compares the above method with parallel or multiple shooting and a method using Chebyshev series. Parallel shooting gives a solution as accurate as that obtained using the Riccati transformation, in a comparable time.     

A second-generation wavelet-based finite element method for the solution of partial differential equations

A new second-generation wavelet (SGW)-based finite element method is proposed for solving partial differential equations (PDEs). An important property of SGWs is that they can be custom designed by selecting appropriate lifting coefficients depending on the application. As a typical problem of SGW algorithm, the calculation of the connection coefficients is described, based on the equivalent filters of SGWs. The formulation of SGW-based finite element equations is derived and a multiscale lifting algorithm for the SGW-based finite element method is developed. Numerical examples demonstrate that the proposed method is an accurate and effective tool for the solution of PDEs, especially ones with singularities.     

The numerical solution of boundary value problems for second order functional differential equations by finite differences

The boundary value problem , 0 <t < 1,x(0)=x(1)=0, is considered. Hereg:R ²R ¹ andF:C[0, 1] C[0, 1]. The solutionx is approximated using finite differences. For a large class of problems it is proved that the approximate solutions exist and converge tox. The method is illustrated by the numerical example.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462, and the Office of Naval Research under Contract No.: N00014-67-A-0128-0004. The computations were supported by the University of Wisconsin Grants Committee.     

Finding numerical solution of linear partial differential equations of first order with real coefficients

Applying Bittner's operational calculus we present a method to give approximate solutions of linear partial differential equations of first order

The numerical solution of parabolic differential equations using variational methods

Summary The question of constructing stable numerical representations for the solutions of initial-boundary value problems for parabolic differential equations is examined.An earlier formulation and discussion of this work can be found in the Author's Ph.D. Thesis (University of Adelaide, South Australia, 1967).     

Modified decomposition solution of nonlinear partial differential equations

Solutions of nonlinear partial differential equations are found using an extended Maclaurin series form of the decomposition method and the Adomian polynomials.     

Hopscotch method: The numerical solution of the Frank-Kamenetskii partial differential equation

Numerical solutions to the Frank-Kamenetskii partial differential equation modelling a thermal explosion in a cylindrical vessel are obtained using the hopscotch scheme. We observe that a nonlinear source term in the equation leads to numerical difficulty and hence adjust the scheme to accommodate such a term. Numerical solutions obtained via MATLAB, MATHEMATICA and the Crank-Nicolson implicit scheme are employed as a means of comparison. To gain insight into the accuracy of the hopscotch scheme the solution is compared to a power series solution obtained via the Lie group method. The numerical solution is also observed to converge to a well-known steady state solution. A linear stability analysis is performed to validate the stability of the results obtained.     

Elliptic systems of partial differential equations and the finite element method

An existence and uniqueness theorem is established for finite element solutions of elliptic systems of partial differential equations. To establish this result, an extension of Gårding's inequality is obtained which is valid for functions that do not necessarily vanish on the boundary of the region. To accomplish this extension, a stronger ellipticity condition, called very strong ellipticity, is defined with various necessary and sufficient conditions given.     

Error analysis of the numerical solution of split differential equations

The operator splitting method is a widely used approach for solving partial differential equations describing physical processes. Its application usually requires the use of certain numerical methods in order to solve the different split sub-problems. The error analysis of such a numerical approach is a complex task. In the present paper we show that an interaction error appears in the numerical solution when an operator splitting procedure is applied together with a lower-order numerical method. The effect of the interaction error is investigated by an analytical study and by numerical experiments made for a test problem.     

Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason