The general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynamics

In this paper, we give the explicit solution to the general Riemann problem for the linearized system of two-dimensional isentropic flow in gas dynamics.     

Front tracking and two-dimensional Riemann problems

A substantial improvement in resolution has been achieved for the computation of jump discontinuities in gas dynamics using the method of front tracking. The essential feature of this method is that a lower dimensional grid is fitted to and follows the discontinuous waves. At the intersection points of these discontinuities, two-dimensional Riemann problems occur. In this paper we study such two-dimensional Riemann problems from both numerical and theoretical points of view. Specifically included is a numerical solution for the Mach reflection, a general classification scheme for two-dimensional elementary waves, and a discussion of problems and conjectures in this area.     

The Hirota equation: Darboux transform of the Riemann–Hilbert problem and higher-order rogue waves

The matrix Riemann–Hilbert problem of the Hirota equation with non-zero boundary conditions is investigated. Based on the matrix Riemann–Hilbert problem, the $n$-fold Darboux transformation is established for the Hirota equation such that $(2n-1,2n)$th-order rogue waves can be found simultaneously. Particularly, we exhibit the first-, second-, third-, and fourth-order rogue waves, and first- and second-order temporal–spatial and spatial periodic breathers for some parameters, respectively.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Riemann problem in non-ideal gas dynamics

In this paper we consider the Riemann problem for gas dynamic equations governing a one dimensional flow of van der Waals gases. The existence and uniqueness of shocks, contact discontinuities, simple wave solutions are discussed using R-H conditions and Lax conditions. The explicit form of solutions for shocks, contact discontinuities and simple waves are derived. The effects of van der Waals parameter on the shock and simple waves are studied. A condition is derived on the initial data for the existence of a solution to the Riemann problem. Moreover, a necessary and sufficient condition is derived on the initial data which gives the information about the existence of a shock wave or a simple wave for a 1-family and a 3-family of characteristics in the solution of the Riemann problem.     

The Riemann solution in the one-dimensional inverse problem

It is shown that the introduction of the Riemann method in potential scattering is a natural device. Thereafter, the one-dimensional Schrödinger equation is considered using the Riemann method. An estimate is found for the associated translation kernel. This estimate reduces to Marchenko's when the domain of the variables is restricted to the positive quadrant of the (x, y) plane. The possibility of a Fourier transform of the Schrödinger equation solution is studied via Riemann's method. Extension of the investigations to the case where a dependence on threshold energies appears in the Schrödinger equation is discussed.     

The torus related Riemann problem

The Riemann jump problem is solved for analytic functions of several complex variables with the unit torus as the jump manifold. A well-posed formulation is given which does not demand any solvability conditions. The higher dimensional Plemelj-Sokhotzki formula for analytic functions in torus domains is established. The canonical functions of the Riemann problem for torus domains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. Thus contrary to earlier research the results are similar to the respective ones for just one variable. A connection between the Riemann and the Riemann-Hilbert boundary value problem for the unit polydisc is explained.     

Canonical form of a two-dimensional Riemann manifold

A proof of the existence on a Riemann manifold of a system of canonical cuts such that the canonical form of the manifold will be a convex region with the minimum number of sides.Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 67–70, January, 1971.The author wishes to thank É. B. Vinberg for suggesting the problem, and A. B. Katk and G. A. Margulis for their comments on the work.     

The Riemann problem for the Leray-Burgers equation

For Riemann data consisting of a single decreasing jump, we find that the Leray regularization captures the correct shock solution of the inviscid Burgers equation. However, for Riemann data consisting of a single increasing jump, the Leray regularization captures an unphysical shock. This behavior can be remedied by considering the behavior of the Leray regularization with initial data consisting of an arbitrary mollification of the Riemann data. As we show, for this case, the Leray regularization captures the correct rarefaction solution of the inviscid Burgers equation. Additionally, we prove the existence and uniqueness of solutions of the Leray-regularized equation for a large class of discontinuous initial data. All of our results make extensive use of a reformulation of the Leray-regularized equation in the Lagrangian reference frame. The results indicate that the regularization works by bending the characteristics of the inviscid Burgers equation and thereby preventing their finite-time crossing.     

分段连续函数为黎曼可积的证法

利用定积分存在的充要条件及其相关性质,给出分段连续函数可积性的两种证明方法.     

The Riemann problem on a finite-sheeted Riemann surface of infinite genus

LetR be the Riemann surface of the functionu(z) specified by the equationu ⁿ=P(z) withn ε ℕ,n ≥ 2, andz ε ℂ, whereP(z) is an entire function with infinitely many simple zeros. OnR, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 25–35, January, 2000.     

Generalised Riemann problem for Euler system

This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves (shock, rarefaction wave and contact discontinuity) is also introduced.     

Unique solvability of the Cauchy problem for the equations of the two-dimensional relativistic chiral fields,taking values in complete Riemann manifolds

One proves the unique global solvability of the Cauchy problem for the two-dimensional quasilinear hyperbolic systems of the theory of chiral fields with values in complete Riemann manifolds.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 81–94, 1981.     

On some generalizations of the classical Riemann problem

The special class of the homogeneous vector boundary Riemann problems on the finite sequence of algebraic surfaces is investigated completely. Its coefficients are the noncommutative permutative matrices of the arbitrary but not prime order, and boundary conditions are given on the system of open contours. The constructive solution procedure and definite structure of the canonical solution matrix are obtained and present some generalizations of the classical Riemann problem. Simultaneously the corresponding class of algebraic equations for the appropriate covering surfaces is formed explicitly too. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)