The stability of difference schemes of second-order of accuracy for hyperbolic-parabolic equations

AbstractA nonlocal boundary value problem for hyperbolic-parabolic equations in a Hilbert space H is considered. Difference schemes of second order of accuracy difference schemes for approximate solution of this problem are presented. Stability estimates for the solution of these difference schemes are established

Numerical methods for solving hyperbolic and parabolic partial differential equations

The main object of this thesis is a study of the numerical 'solution of hyperbolic and parabolic partial differential equations. The introductory chapter deals with a general description and classification of partial differential equations. Some useful mathematical preliminaries and properties of matrices are outlined. Chapters Two and Three are concerned with a general survey of current numerical methods to solve these equations. By employing finite differences, the differential system is replaced by a large matrix system. Important concepts such as convergence, consistency, stability and accuracy are discussed with some detail. The group explicit (GE) methods as developed by Evans and Abdullah on parabolic equations are now applied to first and second order (wave equation) hyperbolic equations in Chapter 4. By coupling existing difference equations to approximate the given hyperbolic equations, new GE schemes are introduced. Their accuracies and truncation errors are studied and their stabilities established. Chapter 5 deals with the application of the GE techniques on some commonly occurring examples possessing variable coefficients such as the parabolic diffusion equations with cylindrical and spherical symmetry. A complicated stability analysis is also carried out to verify the stability, consistency and convergence of the proposed scheme. In Chapter 6 a new iterative alternating group explicit (AGE) method with the fractional splitting strategy is proposed to solve various linear and non-linear hyperbolic and parabolic problems in one dimension. The AGE algorithm with its PR (Peaceman Rachford) and DR (Douglas Rachford) variants is implemented on tridiagonal systems of difference schemes and proved to be stable. Its rate of convergence is governed by the acceleration parameter and with an optimum choice of this parameter, it is found that the accuracy of this method, in general, is better if not comparable to that of the GE class of problems as well as other existing schemes. The work on the AGE algorithm is extended to parabolic problems of two and three space dimensions in Chapter 7. A number of examples are treated and the DR variant is used because of consideration of stability requirement. The thesis ends with a summary and recommendations for future work

Numerical methods for ordinary differential equations with applications to partial differential equations

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.The thesis develops a number of algorithms for the numerical solution of ordinary differential equations with applications to partial differential equations. A general introduction is given; the existence of a unique solution for first order initial value problems and well known methods for analysing stability are described. A family of one-step methods is developed for first order ordinary differential equations. The methods are extrapolated and analysed for use in PECE mode and their theoretical properties, computer implementation and numerical behaviour, are discussed. Lo-stable methods are developed for second order parabolic partial differential equations 1n one space dimension; second and third order accuracy is achieved by a splitting technique in two space dimensions. A number of two-time level difference schemes are developed for first order hyperbolic partial differential equations and the schemes are analysed for Ao-stability and Lo-stability. The schemes are seen to have the advantage that the oscillations which are present with Crank-Nicolson type schemes, do not arise. A family of two-step methods 1S developed for second order periodic initial value problems. The methods are analysed, their error constants and periodicity intervals are calculated. A family of numerical methods is developed for the solution of fourth order parabolic partial differential equations with constant coefficients and variable coefficients and their stability analyses are discussed. The algorithms developed are tested on a variety of problems from the literature.British Governmen

Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit

We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic systems with stiff relaxation in the so-called diffusion limit. In such regime the system relaxes towards a convection-diffusion equation. The first objective of the paper is to show that traditional partitioned IMEX R-K schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system. Of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step. The main goal of the paper is to present an approach, based on IMEX R-K schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the convection-diffusion equation, in which the diffusion is treated implicitly. This is achieved by an original reformulation of the problem, and subsequent application of IMEX R-K schemes to it. An analysis on such schemes to the reformulated problem shows that the schemes reduce to IMEX R-K schemes for the limit equation, under the same conditions derived for hyperbolic relaxation. Several numerical examples including neutron transport equations confirm the theoretical analysis

A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation

In this paper we consider the development of Implicit-Explicit (IMEX) Runge-Kutta schemes for hyperbolic systems with multiscale relaxation. In such systems the scaling depends on an additional parameter which modifies the nature of the asymptotic behavior which can be either hyperbolic or parabolic. Because of the multiple scalings, standard IMEX Runge-Kutta methods for hyperbolic systems with relaxation loose their efficiency and a different approach should be adopted to guarantee asymptotic preservation in stiff regimes. We show that the proposed approach is capable to capture the correct asymptotic limit of the system independently of the scaling used. Several numerical examples confirm our theoretical analysis

Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation

We consider the development of high order space and time numerical methods based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic systems with relaxation. More specifically, we consider hyperbolic balance laws in which the convection and the source term may have very different time and space scales. As a consequence the nature of the asymptotic limit changes completely, passing from a hyperbolic to a parabolic system. From the computational point of view, standard numerical methods designed for the fluid-dynamic scaling of hyperbolic systems with relaxation present several drawbacks and typically lose efficiency in describing the parabolic limit regime. In this work, in the context of Implicit-Explicit linear multistep methods we construct high order space-time discretizations which are able to handle all the different scales and to capture the correct asymptotic behavior, independently from its nature, without time step restrictions imposed by the fast scales. Several numerical examples confirm the theoretical analysis

On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity

The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion.Comment: 7 pages, 3 figure