MULTI-DIMENSIONAL RIEMANN PROBLEM OF SCALAR CONSERVATION LAW

This paper considers multi-dimensional Riemann problem in another kind of view. The author gets solution of (1.1)(1.2) in Theorem 3.4 and proves itu uniqueness. A new method of solution constructing is applied, which is different from the usual self-similar transformation. The author also discusses some generalized concepts in multi-dimensional situation (such as 'convex condition', 'left value' and 'right value', etc). An example is finally given to demonstrate that rarefaction wave solution of (1.1)(1.2) is not self-similar.

THE RIEMANN PROBLEM FOR ISENTROPIC COMPRESSIBLE EULER EQUATIONS WITH DISCONTINUOUS FLUX

We consider the singular Riemann problem for the rectilinear isentropic compressible Euler equations with discontinuous flux,more specifically,for pressureless flow on the left and polytropic flow on the right separated by a discontinuity x=x(t).We prove that this problem admits global Radon measure solutions for all kinds of initial data.The over-compressing condition on the discontinuity x=x(t)is not enough to ensure the uniqueness of the solution.However,there is a unique piecewise smooth solution if one proposes a slip condition on the right-side of the curve x=x(t)+0,in addition to the full adhesion condition on its left-side.As an application,we study a free piston problem with the piston in a tube surrounded initially by uniform pressureless flow and a polytropic gas.In particular,we obtain the existence of a piecewise smooth solution for the motion of the piston between a vacuum and a polytropic gas.This indicates that the singular Riemann problem looks like a control problem in the sense that one could adjust the condition on the discontinuity of the flux to obtain the desired flow field.

A Numerical Study of Riemann Problem Solutions for the Homogeneous One-Dimensional Shallow Water Equations

The solution of the Riemann Problem (RP) for the one-dimensional (1D) non-linear Shallow Water Equations (SWEs) is known to produce four potential wave patterns for the scenario where the water depth is always positive. In this paper, we choose four test problems with exact solutions for the 1D SWEs. Each test problem is a RP with one of the four possible wave patterns as its solution. These problems are numerically solved using schemes from the family of Weighted Essentially Non-Oscillatory (WENO) methods. For comparison purposes, we also include results obtained from the Random Choice Method (RCM). This study has three main objectives. Firstly, we outline the procedures for the implementation of the methods employed in this paper. Secondly, we assess the performance of the schemes in conjunction with a second-order Total Variation Diminishing (TVD) flux on a variety of RPs for the 1D SWEs (for both short- and long-time simulations). Thirdly, we investigate if a single method yields optimal outcomes for all test problems. Optimal outcomes refer to numerical solutions devoid of spurious oscillations, exhibiting high resolution of discontinuities, and attaining high-order accuracy in the smooth parts of the solution.

A branch-and-bound algorithm for multi-dimensional quadratic 0-1 knapsack problems

In this paper, a branch-and-bound method for solving multi-dimensional quadratic 0-1 knapsack problems was studied. The method was based on the Lagrangian relaxation and the surrogate constraint technique for finding feasible solutions. The Lagrangian relaxations were solved with the maximum-flow algorithm and the Lagrangian bounds was determined with the outer approximation method. Computational results show the efficiency of the proposed method for multi-dimensional quadratic 0-1 knapsack problems.

Machine Learning Approaches for the Solution of the Riemann Problem in Fluid Dynamics:a Case Study

We present our results by using a machine learning(ML)approach for the solution of the Riemann problem for the Euler equations of fluid dynamics.The Riemann problem is an initial-value problem with piecewise-constant initial data and it represents a mathematical model of the shock tube.The solution of the Riemann problem is the building block for many numerical algorithms in computational fluid dynamics,such as finite-volume or discontinuous Galerkin methods.Therefore,a fast and accurate approximation of the solution of the Riemann problem and construction of the associated numerical fluxes is of crucial importance.The exact solution of the shock tube problem is fully described by the intermediate pressure and mathematically reduces to finding a solution of a nonlinear equation.Prior to delving into the complexities of ML for the Riemann problem,we consider a much simpler formulation,yet very informative,problem of learning roots of quadratic equations based on their coefficients.We compare two approaches:(i)Gaussian process(GP)regressions,and(ii)neural network(NN)approximations.Among these approaches,NNs prove to be more robust and efficient,although GP can be appreciably more accurate(about 30\%).We then use our experience with the quadratic equation to apply the GP and NN approaches to learn the exact solution of the Riemann problem from the initial data or coefficients of the gas equation of state(EOS).We compare GP and NN approximations in both regression and classification analysis and discuss the potential benefits and drawbacks of the ML approach.

RIEMANN SUMMABILITY OF MULTI-DIMENSIONAL TRIGONOMETRIC-FOURIER SERIES

The d-dimensional classical Hardy spaces H_p (T^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T^d)to L_p(T^2) (d/(d+1)<p≤∞) and is of weak type (1, 1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L_1(T^d)is a.e. Riemann summable to f, provided again that the limit is taken over a positive cone.

THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS

In this article, we study the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtain the global existence of generalized solutions that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data.

ON SOLUTION OF A KIND OF RIEMANN BOUNDARY VALUE PROBLEM WITH SQUARE ROOTS

Solution of the Riemann boundary value problem with square roots (1.1) for analytic functions proposed in [1] is reconsidered, which was solved under certain assumptions on the branch points appeared. Here the work is continued without these assumptions and the problem is solved in the general case.

A Generalized Riemann Boundary Value Problem

The generalized Riemann boundary value problem for analytic functions is considered, where the unknown function may have branch points of the second order. Under certain assumptions, its general solution as well as the condition of solvability is obtained when the solution is required to be of finite order at infinity.

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.

Non-Homogeneous Riemann Boundary Value Problem with Radicals

The solution of the non-homogeneous Riemann boundary value problem with radicals (1. 2) together with its condition of solvability is obtained for arbitrary positive integersp andq, which generalizes the results for the casep=q=2.

RIEMANN BOUNDARY VALUE PROBLEM WITH RESPECT TO THE PERTURBATION OF BOUNDARY CURVE TO BE AN OPEN ARC

By dint of the stability of Cauchy-type integral with kernel density of class H＊ for an open arc, this paper discusses the stability of the solution of Riemann boundary value problem with respect to the perturbation of boundary curve to be an open arc.

TWO-DIMENSIONAL RIEMANN PROBLEMS:FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS

In this paper we survey the authors＇ and related work on two-dimensional Riemann problems for hyperbolic conservation laws, mainly those related to the compressible Euler equations in gas dynamics. It contains four sections： 1. Historical review. 2. Scalar conservation laws. 3. Euler equations. 4. Simplified models.

RIEMANN BOUNDARY VALUE PROBLEMS FOR SOME K-REGULAR FUNCTIONS IN CLIFFORD ANALYSIS

In this paper,we study the R m（m〉0） Riemann boundary value problems for regular functions,harmonic functions and bi-harmonic functions with values in a universal clifford algebra C（Vn,n）.By using Plemelj formula,we get the solutions of R m（m〉0） Riemann boundary value problems for regular functions.Then transforming the Riemann boundary value problems for harmonic functions and bi-harmonic functions into the Riemann boundary value problems for regular functions,we obtain the solutions of R m（m〉0） Riemann boundary value problems for harmonic functions and bi-harmonic functions.

ON HOMOGENEOUS RIEMANN BOUNDARY VALUE PROBLEMS OF HIGHER DEGREE

General solution for homogeneous Riemann problems of higher degree is considered. By introducing the concept of loop as well as cross-point, the problem is solved in detail for the quadratic case. The cubic and the quartic ones are also analysed.

RIEMANN BOUNDARY VALUE PROBLEMS WITH GIVEN PRINCIPAL PART

In this article, the authors discuss the Riemann boundary value problems with given principal part. First, authors consider a special case and give a classification of the solution class Rn by the way. And then, they consider the general case. The solvable conditions for this problem and its solutions is obtained when it is solvable.

A Class of Quasi-Linear Riemann-Hilbert Problems for General Holomorphic Functions in the Unit Disk

In this paper, a class of quasi linear Riemann Hilbert problems for general holomorphic functions in the unit disk was studied. Under suitable hypotheses, the existence of solutions of the Hardy class H 2 to this problem was proved by means of Tikhonov's fixed point theorem and corresponding theories for general holomorphic functions.

Reconsideration on Homogeneous Quadratic Riemann Boundary Value Problem

The homogeneous quadratic riemann boundary value problem (1) with H?lder continuous coefficients for the normal case was considered by the author in 1997. But the solutions obtained there are incomplete. Here its general method of solution is obtained. Key words homogeneous quadratic Riemann boundary value problem - ordinary and special nodes - index - sectionally holomorphic function CLC number O 175.5 Foundation item: Supported by the National Natural Science Foundation of China (19871064)Biography: Lu Jian-ke (1922-), male, Professor, research direction: complex analysis and its applications.

Guckenheimer structure of solution of Riemann problem with four pieces of constants in two space dimensions for scalar conservation laws

By using the generalized characteristic analysis method, the two-dimensional four-wave Riemann problem for scalar conservation laws, which is nonconvex along the y direction, was studied. Riemann solutions, which involve the Guckenheimer structure, were constructed.

Riemann Boundary Value Problem of Non-Normal Type on the Infinite Straight Line

Various kinds of Riemann boundary value problems (BVPs) for analytic functions on closed curves or on open arc, doubly periodic Riemann BVPs, doubly quasi-periodic Riemann BVPs, and BVPs for polyanalytic functions have been widely investigated in [1-8]. The main ap- proach is to use the decomposition of polyanalytic functions and their generalization to transform the boundary value problems to their corresponding boundary value problems for analytic functions. Recently, inverse Riemann BVPs for generalized analytic functions or bianalytic functions have been investigated in [9-12]. In this paper, we consider a kind of Riemann BVP of non-normal type on the infinite straight line and discuss the solvable conditions and the general solution for it.