Homogeneous and non-homogeneous boundary value problems for first order linear hyperbolic systems arising in fluid-mechanics (Part I)

We prove t h e existence and the uniqueness of differentiable and strong solutions for aclass of non-homogeneous boundary value problems for first order linear hyperbolic systems arising from the dynamics of compressible non-viscous fluids . The method provides.the existence of differentiable solutions without resorting to strong or weak solutions. A necessary and sufficient condition for the existence of solutions for the non-homogeneous problem is proved. I t consists of an explicitrelationship between the boundary values of u and those of the data f . Strong solutions are obtained without this supplementary assumption. See Theorems 3.1, 4.1, 4 . 2 , 4.3 and Corollary 4.4; see also Remarks 2.1 and 2.4. In this paper we consider equation (3.1) below. In the forthcoming part II we prove similar results for the corresponding evolution problem.     

Investigation of boundary value problems for elliptic systems of first order

This paper is the author's abstract of a dissertation submitted for the degree of Doctor of Physicomathematical Sciences. The dissertation was defended November 2, 1972 at a session of the Scientific Council of the V. A. Steklov Mathematical Institute of the Academy of Sciences, USSR. Official referees: Corresponding Member of the Academy of Sciences of the USSR Professor A. V. Bitsadze, Corresponding Member of the Academy of Sciences of the USSR Professor M. M. Lavrent'ev, and Doctor of Physicomathematical Sciences, Professor P. I. Lizorkin.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 291–304, August, 1973.     

Existence theorems for linear hyperbolic boundary value problems at resonance

Summary Questions of existence, uniqueness, and continuous dependence for weak solutions of linear hyperbolic boundary value problems are considered. The differential equations have the form u_tt + Au=f, where A is elliptic in the spatial variables, and the boundary conditions are homogeneous in both space and time. Resolution of these questions depends on the relationship of the eigenvalues of A and those of an associated scalar problem in time.     

On the boundary value problem in a dihedral angle for normally hyperbolic systems of first order

Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.     

The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems

Codes for the numerical solution of two-point boundary value problems can now handle quite general problems in a fairly routine and reliable manner. When faced with particularly challenging equations, such as singular perturbation problems, the most efficient codes use a highly non-uniform grid in order to resolve the non-smooth parts of the solution trajectory. This grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control. Similar error estimates are used to decide whether or not to accept a solution. Such an approach is very effective in general providing that the problem to be solved is well conditioned. However, if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution. Even worse, for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance. The primary reason for this is, of course, that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution. In view of this it could be argued that, when solving a two-point boundary value problem in cases where we have no idea of its conditioning, we should provide an estimate of the condition number of the problem as well as the numerical solution. In this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL ₁. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL _p, 1≦p<∞, are given.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Comparison results for first and second order boundary value problems at resonance

Various types of comparison results for first and second order periodic boundary value problems are developed. It is hoped that these comparison results play an important role in the existence theory of boundary value problems at resonance.