Godunov-type Numerical Scheme Research Articles

High-resolution numerical approximation of traffic flow problems with variable lanes and free-flow velocities

This paper develops macroscopic traffic flow models for a highway section with variable lanes and free-flow velocities, that involve spatially varying flux functions. To address this complex physical property, we develop a Riemann solver that derives the exact flux values at the interface of the Riemann problem. Based on this solver, we formulate Godunov-type numerical schemes to solve the traffic flow models. Numerical examples that simulate the traffic flow around a bottleneck that arises from a drop in traffic capacity on the highway section are given to illustrate the efficiency of these schemes.

Simulation of wave and front propagation in thermoelastic materials with phase transformation

A thermodynamically consistent form for the finite-volume numerical method for thermoelastic wave and front propagation is proposed in the paper. Such a reformulation provides the applicability of the Godunov type numerical schemes based on averages of field variables to the description of non-equilibrium situations. The non-equilibrium description is fulfilled by using contact quantities instead of numerical fluxes. The contact quantities satisfy the thermodynamic consistency conditions at interfaces which generalize the classical equilibrium conditions.

AN ENERGETIC RESTRICTION ON A NUMERICAL SCHEME IN ONE-DIMENSIONAL DYNAMIC THERMOELASTICITY

The governing system of equations of the dynamic one-dimensional linear thermoelas ticity is a composed hyperbolic-parabolic system and has been studied little in the literature. A Godunov-type numerical scheme for initial-boundary value problems is constructed. An energy inequality is used to control the stability of the proposed numerical scheme, which also allows us to classify thermoelastic problems in strongly coupled and weakly coupled problems. Most real materials fall into the weakly coupled case when the mechanical power can be neglected in the balance of energy equation. The proposed numerical scheme is used to test that the above property of the weakly coupled case and also to see if, in the strongly coupled case, the solutions of certain initial-boundary value problems show oscillatory behavior, which has certain similarities to those experimentally recorded at very low temperature and thought of as evidence of the existence of the "second sound."

A Godunov-type scheme for relativistic magnetohydrodynamics

The basic properties of relativistic magnetohydrodynamics, as a hyperbolic system of quasi-linear conservation laws, are discussed. These are then used to develop a multidimensional Godunov-type numerical scheme that enforces the magnetic flux conservation. This scheme is based on linear Riemann solvers and has second-order accuracy in smooth regions. The results of thorough test calculations demonstrate that the scheme is robust and can cope with truly ultrarelativistic problems.

On Godunov-Type Schemes for Magnetohydrodynamics: 1. A Model System

In the light of recent analytical results on the MHD Riemann problem, Godunov-type numerical schemes for magnetohydrodynamics (MHD) are revisited. As the first step, a model system that exactly preserves the MHD hyperbolic singularities is considered. For this model, analytical results on shock waves are summarized and critical problems occurring in developing shock-capturing methods are identified. Using the results, we propose a new way to define fluxes on cell interfaces. It consists of two solvers, one on the well-posed Riemann problem and another on the evolution of Alfvén waves. Numerical experiments show that the new scheme is more efficient in calculating large-time solutions.

First and Second-Order Accurate Schemes for Two-Fluid Models

The six-equation two-fluid model of two-phase flow taken from the RELAP5/MOD3 computer code has been used to simulate three simple transients: a two-phase shock tube problem, the Edwards Pipe experiment, and water hammer due to rapid valve closure. These transients can be characterized as fast transients, since their characteristic time-scales are determined by the sonic velocity. First and second-order accurate numerical methods have been applied both based on the well-known, Godunov-type numerical schemes. Regarding the uncertainty of the two-fluid models in today’s large computer codes for the nuclear thermal-hydraulics, use of second-order schemes is not always justified. While this paper shows the obvious advantage of second-order schemes in the area of fast transients, first-order accurate schemes may still be sufficient for a wide range of two-phase flow transients where the convection terms play a minor role compared to the source terms.

Method for Noise Suppressing Nozzle Calculation and First Results of Its Implementation

A description of the method for the e owe eld in the noise suppressing nozzle calculation is presented. The method is formulated for the Favre-averaged Navier ‐ Stokes equations. It is based on an explicit monotone second-order approximation, Godunov-type numerical scheme. The e rst set of calculations is made without turbulent and molecular viscosity being taken into account and shows that the Euler approach allows the main features of the complicated e ow in the mixer ‐ ejector nozzle to be described. Special experiments cone rm this idea and make the exactness of computational e uid dynamics results more clear.

Front Tracking for Shear Bands in an Antiplane Shear Model

In this paper we describe a numerical algorithm for the study of shear band, formation and growth in two-dimensional antiplane shear. The constitutive model uses a non-associative flow rule. The numerical scheme is based on a Godunov method for updating the velocity, while the stress is updated via integration along particle paths. The scheme is coupled with a front tracking algorithm for careful evolution of the shear bands. The main challenges are the non-hyperbolicity of the shear band formation and growth (front tracking avoids the catastrophic effects of the loss of hyperbolicity in the Godunov-type numerical scheme), the existence of endpoints for the shear band (the tracked front does not separate the computational domain into disconnected regions), and the non-hyperbolic rate of growth of the shear band. We give examples of the success of the algorithm and show convergence tests.

Approximate Riemann solvers and waves on a nonlinear elastic string

If variable boundary conditions are imposed on a stretched nonlinear elastic string, elementary methods of solution fail, in general. If shocks develope, characteristic theory cannot be used after the shock forms. In the present note we examine the behaviour of the resultant wave motion using Godunov type numerical schemes with various Riemann solvers. Approximate Riemann solvers of the form suggested by Harten, Lax, and Van Leer [6] are compared with an exact solver. Up to the first breakdown time, the solutions may be compared to the solution obtained from characteristic theory.