Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday

December 9,2022 is the 90th birthday of Tong Zhang,a mathematician in Institute of Mathematics,Chinese Academy of Sciences where he was always working on the Riemann problem for gas dynamics in his mathematical life.To celebrate his 90th birthday and great contributions to this specifc feld,we organize this focused issue in the journal Communications on Applied Mathematics and Computation,since the Riemann problem has been proven to be a building block in all felds of theory,numerics and applications of hyperbolic conservation laws.

A New Differential Operator Method to Study the Mechanical Vibration

In this paper, we propose a unified differential operator method to study mechanical vibrations, solving inhomogeneous linear ordinary differential equations with constant coefficients. The main advantage of this new method is that the differential operator D in the numerator of the fraction has no effect on input functions (i.e., the derivative operation is removed) because we take the fraction as a whole part in the partial fraction expansion. The method in various variants is widely implemented in related fields in mechanics and engineering. We also point out that the same mistakes in the differential operator method are found in the related references [1-4].

Two-Dimensional Regular Shock Reflection for the Pressure Gradient System of Conservation Laws

We establish the existence of a global solution to a regular reflection of a shock hitting a ramp for the pressure gradient system of equations. The set-up of the reflection is the same as that of Mach＇s experiment for the compressible Euler system, i.e., a straight shock hitting a ramp. We assume that the angle of the ramp is close to 90 degrees. The solution has a reflected bow shock wave, called the diffraction of the planar shock at the compressive corner, which is mathematically regarded as a free boundary in the self-similar variable plane. The pressure gradient system of three equations is a subsystem, and an approximation, of the full Euler system, and we offer a couple of derivations.

A Global Solution to a Two-dimensional Riemann Problem Involving Shocks as Free Boundaries

We present a global solution to a Riemann problem for the pressure gradient system of equations. The Riemann problem has initially two shock waves and two contact discontinuities. The angle between the two shock waves is set initially to be close to 180 degrees. The solution has a shock wave that is usually regarded as a free boundary in the self-similar variable plane. Our main contribution in methodology is handling the tangential oblique derivative boundary values.

The Riemann Problem for a Blood Flow Model in Arteries

In this paper,the Riemann solutions of a reduced 6×6 blood flow model in mediumsized to large vessels are constructed.The model is nonstrictly hyperbolic and non-conservative in nature,which brings two difficulties of the Riemann problem.One is the appearance of resonance while the other one is loss of uniqueness.The elementary waves include shock wave,rarefaction wave,contact discontinuity and stationary wave.The stationary wave is obtained by solving a steady equation.We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data.We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution.The Riemann solutions may contribute to the design of numerical schemes,which can apply to the complex blood flows.

Two-Dimensional Riemann Problems for the Compressible Euler System

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult,but a viable alternative remains missing.The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems,to illustrate their worthiness.Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics;that is,simplified model is built via asymptotic analysis,numerical simulation and theoretical analysis.A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process.State-of-the-art numerical methods in numerical simulations are used to discern small-scale structures of the solutions,e.g.,semi-hyperbolic patches.Analytical methods are used to establish the validity of the structure revealed in the numerical simulation.The entire process,used in many of Majda's programs,is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws.