Riemann Problem Research Articles

Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics

We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative l2 error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference “ground truth”, and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.

Gas-kinetic scheme for partially ionized plasma in hydrodynamic regime

Most plasmas are only partially ionized. To better understand the dynamics of these plasmas, the behaviors of a mixture of neutral species and plasma in ideal magnetohydrodynamic states are investigated. The current approach is about the construction of coupled kinetic models for the neutral gas, electron, and proton, and the development of the corresponding gas-kinetic scheme (GKS) for the solution in the continuum flow regime. The scheme is validated in the 1D Riemann problem for an enlarged system with the interaction from the Euler waves of the neutral gas and magnetohydrodynamic ones of the plasma. Additionally, the Orszag–Tang vortex problem across different ionized states is tested to examine the influence of neutrals on the MHD wave evolution. These tests demonstrate that the proposed scheme can capture the fundamental features of ideal partially ionized plasma, and a transition in the wave structure from the ideal MHD solution of the fully ionized plasma to the Euler solution of the neutral gas is obtained.

Riemann problem solutions for a balance law under Dirac-Delta source with a discontinuous flux

We study the Riemann problem for a new model on immiscible vertical two-phase flow under point injection. The point injection is modeled by a Dirac [Formula: see text]-source term as well as by a spatially discontinuous flux function, which defines two fluxes, one on each side of the discontinuity. The solutions comprise up to three wave groups: downward waves, a stationary shock and upward waves. Owning the interplay between the Dirac [Formula: see text]-source and the discontinuous flux, there is no standard entropy condition for the stationary shock (flux’s connections). An entropy condition was deduced based on impinging characteristics and perturbation of the constant solution. This condition leads to shocks that do not satisfy the classical Lax’s conditions — even for arbitrarily small shocks — and may also have no viscous profile. The Rankine–Hugoniot condition — embedding the Dirac [Formula: see text]-source — and the entropy condition are geometrically represented by “Flux Projections” that support the analytical method proposed in this paper. We then obtain a [Formula: see text] analytic solution for all initial value problems. We verify the entropy condition using a Lagrangian–Eulerian scheme recently introduced in the literature, which is based in the new concept of no-flow curves. Analytic and numerical solutions fit.

Enhancing accuracy and efficiency: A novel implicit–explicit approach for fluid dynamics simulation

This study presents an innovative implicit–explicit time-stepping algorithm based on a first-order temporal accuracy method, addressing challenges in simulating all-regimes of fluid flows. The algorithm's primary focus is on mitigating stiffness inherent in the density-based “Roe” method, pivotal in finite volume approaches employing unstructured meshes. The objective is to comprehensively evaluate the method's efficiency and robustness, contrasting it with the explicit fourth-order Runge–Kutta method. This evaluation encompasses simulations across a broad spectrum of Mach numbers, including scenarios of incompressible and compressible flow. The scenarios investigated include the Sod Riemann problem to simulate compressible Euler equations, revealing the algorithm's versatility, and the low Mach number Riemann problem to analyze system stiffness in incompressible flow. Additionally, Navier–Stokes equations are employed to study viscous and unsteady flow patterns around stationary cylinders. The study scrutinizes two time-stepping algorithms, emphasizing accuracy, stability, and computational efficiency. The results demonstrate the implicit–explicit Runge–Kutta algorithm's superior accuracy in predicting flow discontinuities in compressible flow. This advantage arises from the semi-implicit nature of the equations, reducing numerical errors. The algorithm significantly enhances accuracy and stability for low Mach number Riemann problems, addressing increasing stiffness as Mach numbers decrease. Notably, the algorithm optimizes computational efficiency for both low Mach number Riemann problems and viscous flows around cylinders, reducing computational costs by 38%–68%. The investigation extends to a two dimensional hypersonic inviscid flow over cylinder and double Mach reflection case, showcasing the method's proficiency in capturing complex and hypersonic flow behavior. Overall, this research advances the understanding of time discretization techniques in computational fluid dynamics, offering an effective approach for handling a wide range of Mach numbers while improving accuracy and efficiency.

The Riemann problem for the Chaplygin gas dynamics with a single-point heating source

This paper is devoted to the Riemann solutions to the Chaplygin gas dynamics with a single-point heating source. To deal with Dirac measure source term, we transform this model into a system of conservation laws and then present a definition of solution to this system of conservation laws. Based on this definition, we establish, respectively, the condition for the occurrence of stationary contact discontinuity solution and delta standing wave solution. With the help of these results, by analyzing the possible combination of various waves, ten kinds of exact solutions and the criteria for the emergence of each solution are established. It is observed that a delta standing wave solution arises for certain initial values, where two components of this solution contain the Dirac measure. It is also noticed that a stationary contact discontinuity followed by a delta shock wave emerges in some solutions. Moreover, these solutions can be used to test the validation of numerical algorithm for a system of conservation laws with a singular source term.

The concentration and cavitation in the Riemann solution for non-homogeneous logrithmic equation of state with magnetic field

In the present study, the phenomena of concentration and cavitation in the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field is analyzed. Firstly, we introduced new state variable for the velocity to modify the non-conservative system into conservative system and solved the Riemann problem for modified system constructively. Further, the Riemann solutions for the transport equations is investigated as pressure and magnetic field vanish. It is proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two shock waves converges to the delta shock wave solution of the transport equations as pressure and magnetic field vanish. It is also proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two rarefaction waves converges to the contact discontinuity solution of the transport equations as pressure and magnetic field vanish.

Interactions between elementary waves and weak discontinuity in two-layer blood flow through artery

Blood flow through arterial circulation can be characterized by fluid transport in flexible tubes and modeled mathematically using the conservation of mass and momentum. A one-dimensional model for two-layer blood flow with different blood velocities and the same constant density in each layer derived from the Euler equations of gas dynamics by taking the vertical average across each layer. This work presents interactions of elementary waves with a weak discontinuity for the quasilinear 3 × 3 system of conservation laws governing the two-layer blood flow in arteries. Exploiting elementary waves as a single-parameter curve, we study the Riemann solution uniquely and consequently establish the condition on initial data for the existence of a solution to the Riemann problem. Furthermore, we discuss the evolution of weak discontinuity waves and subsequently derive their amplitudes; in what follows, we investigate the interactions of weak discontinuity with contact discontinuity and shocks. Finally, a series of numerical tests have been performed to understand the impact of shock strength and the initial data on the amplitudes of reflected and transmitted waves and the jumps in shock acceleration.

Mesoscopic Kinetic Approach of Nonequilibrium Effects for Shock Waves.

A shock wave is a flow phenomenon that needs to be considered in the development of high-speed aircraft and engines. The traditional computational fluid dynamics (CFD) method describes it from the perspective of macroscopic variables, such as the Mach number, pressure, density, and temperature. The thickness of the shock wave is close to the level of the molecular free path, and molecular motion has a strong influence on the shock wave. According to the analysis of the Chapman-Enskog approach, the nonequilibrium effect is the source term that causes the fluid system to deviate from the equilibrium state. The nonequilibrium effect can be used to obtain a description of the physical characteristics of shock waves that are different from the macroscopic variables. The basic idea of the nonequilibrium effect approach is to obtain the nonequilibrium moment of the molecular velocity distribution function by solving the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann BGK) equations or multiple relaxation times Boltzmann (MRT-Boltzmann) equations and to explore the nonequilibrium effect near the shock wave from the molecular motion level. This article introduces the theory and understanding of the nonequilibrium effect approach and reviews the research progress of nonequilibrium behavior in shock-related flow phenomena. The role of nonequilibrium moments played on the macroscopic governing equations of fluids is discussed, the physical meaning of nonequilibrium moments is given from the perspective of molecular motion, and the relationship between nonequilibrium moments and equilibrium moments is analyzed. Studies on the nonequilibrium effects of shock problems, such as the Riemann problem, shock reflection, shock wave/boundary layer interaction, and detonation wave, are introduced. It reveals the nonequilibrium behavior of the shock wave from the mesoscopic level, which is different from the traditional macro perspective and shows the application potential of the mesoscopic kinetic approach of the nonequilibrium effect in the shock problem.

Shock Wave Evaluation in Shell and Tube Heat Exchanger Tube Rupture Scenario

Abstract Shell and Tube Heat Exchangers (STHEs) are commonly used in oil and gas plants to cool or heat process fluids. A possible issue when designing such systems is represented by the tube rupture scenario, when the internal HP (high-pressure) fluid is suddenly discharged to the LP (low-pressure) fluid in the shell. Since, usually, the tube and shell sides have different design pressures, this scenario must be analyzed to assess if the safety measures are fit to protect the weakest part of the system. The time evolution of this event is characterized by different phases; in the first one, a shock wave is generated and propagated very rapidly in the system, with a time scale in the order of a few ms. The phenomenon is so rapid that pressure relief devices are not effective. This kind of wave is normally not evaluated in the majority of available publications; however, it is deemed that it is of particular relevance for large-size SHTEs, due to their significant investment cost, and hence, a methodology has been developed in this work for the purpose. The main impact of such waves is expected in the LP piping connected to the exchanger. The amplitude of the shock wave at the source location, i.e., at the ruptured tube, is calculated based on the theory of the Riemann problem, with reference to different types of HP and LP fluids. Then, the relevant propagation up to the piping entrance is studied to estimate the wave damping and the effective wave amplitude impacting it. Both validation calculations and calculations referred to new design applications and existing installations are presented, and possible mitigation measures are proposed, if any.

Novel hydrodynamic schemes capturing shocks and contact discontinuities and comparison study with existing methods

We present a new hydrodynamic scheme named Godunov Density-Independent Smoothed Particle Hydrodynamics (GDISPH), that can accurately handle shock waves and contact discontinuities without any manually tuned parameters. This is in contrast to the standard formulation of smoothed particle hydrodynamics (SSPH), which requires the parameters for an artificial viscosity term to handle the shocks and struggles to accurately handle the contact discontinuities due to unphysical repulsive forces, resulting in surface tension that disrupts pressure equilibrium and suppresses fluid instabilities. While Godunov SPH (GSPH) can handle the shocks without the parameters by using solutions from a Riemann solver, it still cannot fully handle the contact discontinuities. Density-Independent Smoothed Particle Hydrodynamics (DISPH), one of several schemes proposed to handle contact discontinuities more effectively than SSPH, demonstrates superior performance in our tests involving strong shocks and contact discontinuities. However, DISPH still requires the artificial viscosity term. We integrate the Riemann solver into DISPH in several ways, yielding some patterns of GDISPH. The results of standard tests such as the one-dimensional Riemann problem, pressure equilibrium, Sedov–Taylor, and Kelvin–Helmholtz tests are favourable to GDISPH Case 1 and GDISPH Case 2, as well as DISPH. We conclude that GDISPH Case 1 has an advantage over GDISPH Case 2effectively handling shocks and contact discontinuities without the need for specific parameters or introducing any additional numerical diffusion.

The initial value problem of coupled Aw-Rascle traffic model with Chaplygin pressure

This paper studies the initial value problem of coupled Aw-Rascle (CAR) traffic model with Chaplygin pressure. A definition of weak solution to CAR model is developed to obtain the waves with phase transition. Combining the waves of Aw-Rascle traffic model, we construct the solutions to the Riemann problem of CAR model with Chaplygin pressure. Without imposing additional coupling conditions, we obtain the existence, uniqueness and stability of Riemann solutions. Then we study the interaction problem of the CAR model with Chaplygin pressure, which models traffic phenomenon with careless driving in a certain section of the highway. From the solutions to the interaction problem, we can observe the formation and disappearance of traffic jam. The mathematical mechanism of traffic jam is analyzed, and the result is in good agreement with real traffic flow.

Efficient non-iterative multi-point method for solving the Riemann problem

Efficient non-iterative multi-point method for solving the Riemann problem

A Numerical Method for Cross-Section Changes and Gates Resulting from the Riemann Problem in Irrigation Canal Systems

A Numerical Method for Cross-Section Changes and Gates Resulting from the Riemann Problem in Irrigation Canal Systems

Numerical simulation of water jet load induced by spherical bubble collapse under underwater explosion

In this study, a numerical method combining the HLLC solver and FV-WENO scheme is proposed to accurately simulate the behavior of multiphase compressible fluids and investigate the formation process of water jets. The HLLC solver approximates the conservation equations of the fluid by using numerical fluxes from the approximate Riemann problem, while the FV-WENO scheme improves the accuracy of the numerical solution by employing the weighted essentially non-oscillatory (WENO) reconstruction method. The combination of these methods allows for a more accurate simulation of the behavior of multiphase compressible fluids, leading to an accurate depiction of the formation process of water jets. By simulating the interaction between underwater shock waves from explosions and single or multiple spherical bubbles, we study the generation, development, and collapse of water jets during the bubble collapse process. By revealing the characteristics of the load during the formation of water jets and the interaction mechanisms between bubbles, our study provides support for further research on water jet phenomena. The formation process of water jets is of great significance for understanding the propagation of underwater shock waves and structural damage. Through this study, we delve into the mechanism of water jet formation, providing a new perspective on the propagation of underwater shock waves and structural damage. Our research results demonstrate that the application of the HLLC solver and FVWENO scheme in water jet simulation effectively improves the accuracy and precision of the numerical solution.

The limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force

In this paper, we investigate the limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for modified Chaplygin gas with the body force as the two parameters tend to zero. The formation of delta shock waves and the vacuum states is identified and analyzed during the process of vanishing pressure in the Riemann solutions. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, our main objective is to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force is solved. Second, we rigorously confirm that, as the pressure vanishes, any two shock Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a δ-shock solution to the pressureless gas dynamics model with a body force, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.

Comparison of Computational Methods and Approaches Applied in Formulation of Boundary Conditions in Lagrange’s Ballistic Problem

This paper compares algorithms for constructing a differential solution to the gas dynamics equations in the surrounding of a moving boundary. In order to compare the algorithms, the Lagrange problem, also known as the piston problem, was chosen as a test problem. A comparison was made between the author’s algorithm based on the method of characteristics and the classical approach with a fictitious cell using the solution of the Riemann problem. Calculations were carried out for the case of subsonic and supersonic gas motion. Comparisons were made on a fixed grid with a dynamically expanding cell (Euler grid) and on a uniformly expanding grid (Arbitrary Lagrangian-Eulerian – ALE grid). The problem was solved using numerical schemes of second order in time and space Lax - Wendroff type: the Richtmyer scheme and the MacCormack scheme. The results of calculations using the characteristics method were used as a benchmark solution. The comparative analysis carried out allows conclusions to be drawn regarding the accuracy of the individual approaches, as well as the scope of their applicability.

Glimm’s method and density of wild data for the Euler system of gas dynamics *

We adapt Glimm’s approximation method to the framework of convex integration to show density of wild data for the (complete) Euler system of gas dynamics. The desired infinite family of entropy admissible solutions emanating from the same initial data is obtained via convex integration of suitable Riemann problems pasted with local smooth solutions. In addition, the wild data belong to BV class.

Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

Convergence of a Generalized Riemann Problem Scheme for the Burgers Equation

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

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

Numerical viscosity control in Godunov-like smoothed particle hydrodynamics for realistic flows modeling

In this study, we introduce a way to control the viscosity of the numerical approximation in the Godunov-like smoothed particle hydrodynamics (SPH) methods. This group of SPH methods includes momentum and energy fluxes in the right-hand sides of the equations, which are calculated by the solution of the Riemann problem between each pair of neighboring particles within the support radius of the smoothing kernel, which is similar to the procedure for the calculation of fluxes across cell boundaries in Godunov schemes. Such SPH methods do not require the use of artificial viscosity since the significant numerical viscosity is already introduced by a Riemann problem solution. We demonstrate that such a numerical viscosity may be measured and obtain the explicit expression for it depending on smoothed particle properties. In particular, we have found that Godunov-like SPH method with interparticle contact algorithms produces numerical viscosity several orders of magnitude higher than physical viscosity in materials. Modern approaches, such as SPH with monotonic upstream-centered scheme for conservation laws or weighted essentially non-oscillatory reconstruction techniques, have not only lower numerical viscosity but also too large for modeling real-world viscous flows. By constructing a correcting viscous stress tensor based on the analytical solution for discontinuous viscous flow, it is possible to reduce the viscous stresses of numerical origin. The use of such a correction makes it possible to improve the agreement with experiments in the simulation of viscous flows without using schemes of higher order reconstruction.