Value Problem For Hyperbolic Systems Research Articles

Method of solution of certain boundary value problems for hyperbolic systems of quasilinear equations of first order with two variables: PMM vol. 39, n≗2, 1975, pp. 253–259

We propose a method of obtaining exact solutions of certain boundary value problems for hyperbolic systems of quasilinear equations of first order with two unknowns. The method utilizes special series. As an example, we solve the problem of motion of a plane, cylindrical or spherical piston in a gas with distributed density.

Extrapolation applied to the method of characteristics for a first order system of two partial differential equations

The proceduresivp presented in this paper calculates an approximate solution of Cauchy's initial value problem for hyperbolic systems of the form (1) (see below). The discretization which proceeds along the characteristics is performed using the midpoint rule started by Euler's method. To provide an algorithm of high accuracy the numerical solution is improved by step size extrapolation. This paper contains anAlgol program completed by examples of the use and test results.

The Convergence Rate for Difference Approximations to Mixed Initial Boundary Value Problems

The convergence rate for difference approximations to mixed initial boundary value problems for hyperbolic systems is treated. Assuming that the approximation at the boundary has one-order lower accuracy than at inner points, conditions are given such that the overall accuracy of the solution is kept at the higher order.

The convergence rate for difference approximations to mixed initial boundary value problems

The convergence rate for difference approximations to mixed initial boundary value problems for hyperbolic systems is treated. Assuming that the approximation at the boundary has one-order lower accuracy than at inner points, conditions are given such that the overall accuracy of the solution is kept at the higher order.

Initial-Boundary Value Problems for Hyperbolic Systems in Regions with Corners. II

In the previous paper in this series we obtained conditions equivalent to the validity of certain energy estimates for a general class of hyperbolic systems in regions with corners. In this paper we examine closely the phenomena which occur near the corners if these conditions are violated. These phenomena include: the development of strong singularities (lack of existence), travelling waves which pass unnoticed through the corner (lack of uniqueness), existence and uniqueness if and only if additional conditions are imposed at the corner, and weak solutions which are not strong solutions. We also systematically analyze the conditions for certain important problems. We discuss the physical and computational significance of these results.

Initial-boundary value problems for hyperbolic systems in regions with corners. II

In the previous paper in this series we obtained conditions equivalent to the validity of certain energy estimates for a general class of hyperbolic systems in regions with corners. In this paper we examine closely the phenomena which occur near the corners if these conditions are violated. These phenomena include: the development of strong singularities (lack of existence), travelling waves which pass unnoticed through the corner (lack of uniqueness), existence and uniqueness if and only if additional conditions are imposed at the corner, and weak solutions which are not strong solutions. We also systematically analyze the conditions for certain important problems. We discuss the physical and computational significance of these results.

Initial-Boundary Value Problems for Hyperbolic Systems in Regions with Corners. I

In recent papers Kreiss and others have shown that initial-boundary value problems for strictly hyperbolic systems in regions with smooth boundaries are well-posed under uniform LopatinskiÄ­ conditions. In the present paper the author obtains new conditions which are necessary for existence and sufficient for uniqueness and for certain energy estimates to be valid for such equations in regions with corners. The key tool is the construction of a symmetrizer which satisfies an operator valued differential equation.

Initial-boundary value problems for hyperbolic systems in regions with corners. I

In recent papers Kreiss and others have shown that initial-boundary value problems for strictly hyperbolic systems in regions with smooth boundaries are well-posed under uniform LopatinskiÄ­ conditions. In the present paper the author obtains new conditions which are necessary for existence and sufficient for uniqueness and for certain energy estimates to be valid for such equations in regions with corners. The key tool is the construction of a symmetrizer which satisfies an operator valued differential equation.

Initial boundary value problems for hyperbolic systems

Initial boundary value problems for hyperbolic systems