Hyperbolic Systems Of Conservation Laws Research Articles

Roe‐type Riemann solver for gas–liquid flows using drift‐flux model with an approximate form of the Jacobian matrix

SummaryThis work presents an approximate Riemann solver to the transient isothermal drift‐flux model. The set of equations constitutes a non‐linear hyperbolic system of conservation laws in one space dimension. The elements of the Jacobian matrix A are expressed through exact analytical expressions. It is also proposed a simplified form of A considering the square of the gas to liquid sound velocity ratio much lower than one. This approximation aims to express the eigenvalues through simpler algebraic expressions. A numerical method based on the Gudunov's fluxes is proposed employing an upwind and a high order scheme. The Roe linearization is applied to the simplified form of A. The proposed solver is validated against three benchmark solutions and two experimental pipe flow data. Copyright © 2015 John Wiley & Sons, Ltd.

Comparison study of the reactive and predictive dynamic models for pedestrian flow

This paper formulates the reactive and predictive dynamic models for pedestrian flow and presents a comparison of the two models. The path-choice behavior of pedestrians in the reactive dynamic model is described that pedestrians tend to walk along a path with the lowest instantaneous cost. The desired walking direction of pedestrians in the predictive dynamic model is chosen to minimize the actual cost based on predictive traffic conditions. An algorithm used to solve the two models encompasses a cell-centered finite volume method for a hyperbolic system of conservation laws and a time-dependent Hamilton–Jacobi equation, a fast sweeping method for an Eikonal-type equation, and a self-adaptive method of successive averages for an arisen discrete fixed point problem. The two models and their algorithm are applied to investigate the spatio-temporal patterns of flux or density and path-choice behaviors of pedestrian flow marching in a facility scattered with an obstacle. Numerical results show that the two models are able to capture macroscopic features of pedestrian flow, traffic instability and other complex nonlinear phenomena in pedestrian traffic, such as the formation of stop-and-go waves and clogging at bottlenecks. Different path-choice strategies of pedestrians cause different spatial distributions of pedestrian density specially in the high-density regions (near the obstacle and exits).

Long-time Stability in Systems of Conservation Laws, Using Relative Entropy/Energy

We study the long-time stability of shock-free solutions of hyperbolic systems of conservation laws, under an arbitrarily large initial disturbance in L2∩ L∞. We use the relative entropy method, a robust tool which allows us to consider rough and large disturbances. We display practical examples in several space dimensions, for scalar equations as well as isentropic gas dynamics. For full gas dynamics, we use a trick from Chen [1], in which the estimate is made in terms of the relative mechanical energy instead of the relative mathematical entropy.

Generalized solution to a system of conservation laws which is not strictly hyperbolic

In this paper we study a non-strictly hyperbolic system of conservation laws when viscosity is present and when viscosity is zero, which has been studied in [12]. We show the existence and uniqueness of the solution in the space of generalized functions of Colombeau for the viscous problem and construct a solution to the inviscid system in the sense of association. Also we construct a solution using shadow wave approach [17] and Volpert product which was partially determined as vanishing viscosity limit in [12].

Weak–strong hyperbolic splitting for simulating conservation laws

A simple and accurate central scheme in finite volume framework is developed for systems of hyperbolic conservation laws, using a splitting of strongly hyperbolic and weakly hyperbolic parts. This leads to the flux function of 1D inviscid Euler compressible system being split into convection and pressure parts and 1D inviscid shallow water system into convection and celerity parts. The numerical diffusion is fixed based on flux equivalence principle, which leads to the satisfaction of the jump conditions. The numerical scheme is tested on various shock tube problems of gas dynamics for 1D Euler equations and on dam breaking problems for shallow water equations. Comparison is done with an approximate Riemann solver to demonstrate the efficiency of the numerical method.

Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws

We study analytical properties of the Toro-Titarev solver for generalized Riemann problems (GRPs), which is the heart of the flux computation in ADER generalized Godunov schemes. In particular, we compare the Toro-Titarev solver with a local asymptotic expansion developed by LeFloch and Raviart. We show that for scalar problems the Toro-Titarev solver reproduces the truncated Taylor series expansion of LeFloch-Raviart exactly, whereas for nonlinear systems the Toro-Titarev solver introduces an error whose size depends on the height of the jump in the initial data. Thereby, our analysis answers open questions concerning the justification of simplifying steps in the Toro-Titarev solver. We illustrate our results by giving the full analysis for a nonlinear 2-by-2 system and numerical results for shallow water equations.

Central finite volume schemes on nonuniform grids and applications

We propose a new one-dimensional unstaggered central scheme on nonuniform grids for the numerical solution of homogeneous hyperbolic systems of conservation laws with applications in two-phase flows and in hydrodynamics with and without gravitational effect. The numerical base scheme is a generalization of the original Lax–Friedrichs scheme and an extension of the Nessyahu and Tadmor central scheme to the case of nonuniform irregular grids. The main feature that characterizes the proposed scheme is its simplicity and versatility. In fact, the developed scheme evolves a piecewise linear numerical solution defined at the cell centers of a nonuniform grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of staggered cells used intermediately. Spurious oscillations are avoided using a slopes limiting procedure. The developed scheme is then validated and used to solve classical problems arising in gas–solid two phase flow problems. The proposed scheme is then extended to the case of non-homogenous hyperbolic systems with a source term, in particular to the case of Euler equations with a gravitational source term. The obtained numerical results are in perfect agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and potential of the proposed method to handle both homogeneous and non-homogeneous hyperbolic systems.

Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws

AbstractAn entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper to approximate hyperbolic systems of conservation laws. This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. In this paper, we design efficient preconditioners for the large, non-symmetric linear system, that needs to be solved at every Newton step. Two sets of preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are designed. Fourier analysis of the preconditioners reveals their robustness and a large number of numerical experiments are presented to illustrate the gain in efficiency that results from preconditioning. The resulting method is employed to compute approximate solutions of the compressible Euler equations, even for very high CFL numbers.

Solution of multi-component, two-phase Riemann problems with constant pressure boundaries

Riemann problems for two associated hyperbolic systems of conservation laws are considered. The Riemann problem for constant flow velocity finds existing solutions in the literature. Here, it is proved that the associated Riemann problem with the alternative assumption of constant pressure boundaries can be calculated from the constant velocity solution. This introduces the total velocity as an unknown function of time, which is explicitly determined in an algorithmic fashion.

Tikhonov theorem for linear hyperbolic systems

A class of linear hyperbolic systems of conservation laws with multiple time scales which are modeled by a perturbation parameter is considered in this paper. By setting the perturbation parameter to zero, two subsystems, the reduced subsystem standing for the slow dynamics and the boundary-layer subsystem representing the fast dynamics, are computed. It is first proved that the exponential stability of the full system implies the stability of both subsystems. Secondly, a counterexample is given to indicate that the stability of the two subsystems does not ensure the full system’s stability. Moreover a new Tikhonov theorem for this class of infinite dimensional systems is stated. The solution of the full system can be approximated by that of the reduced subsystem, and this is proved by Lyapunov techniques. An application to boundary feedback stabilization of gas transport model is used to illustrate the results.

Direct Arbitrary-Lagrangian–Eulerian ADER-MOOD finite volume schemes for multidimensional hyperbolic conservation laws

In this paper we present a new family of efficient high order accurate direct Arbitrary-Lagrangian–Eulerian (ALE) one-step ADER-MOOD finite volume schemes for the solution of nonlinear hyperbolic systems of conservation laws for moving unstructured triangular and tetrahedral meshes. This family is the next generation of the ALE ADER-WENO schemes presented in [16,20]. Here, we use again an element-local space–time Galerkin finite element predictor method to achieve a high order accurate one-step time discretization, while the somewhat expensive WENO approach on moving meshes, used to obtain high order of accuracy in space, is replaced by an a posteriori MOOD loop which is shown to be less expensive but still as accurate. This a posteriori MOOD loop ensures the numerical solution in each cell at any discrete time level to fulfill a set of user-defined detection criteria. If a cell average does not satisfy the detection criteria, then the solution is locally re-computed by progressively decrementing the order of the polynomial reconstruction, following a so-called cascade of predefined schemes with decreasing approximation order. A so-called parachute scheme, typically a very robust first order Godunov-type finite volume method, is employed as a last resort for highly problematic cells. The cascade of schemes defines how the decrementing process is carried out, i.e. how many schemes are tried and which orders are adopted for the polynomial reconstructions. The cascade and the parachute scheme are choices of the user or the code developer. Consequently the iterative MOOD loop allows the numerical solution to maintain some interesting properties such as positivity, mesh validity, etc., which are otherwise difficult to ensure. We have applied our new high order unstructured direct ALE ADER-MOOD schemes to the multi-dimensional Euler equations of compressible gas dynamics. A large set of test problems has been simulated and analyzed to assess the validity of our approach in terms of both accuracy and efficiency (CPU time and memory consumption).

Stability and convergence of relaxation schemes to hyperbolic balance laws via a wave operator

This paper deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [Hyperbolic systems of balance laws with weak dissipation, J. Hyp. Diff. Equations 3 (2006) 505–527.], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented by [Ch. Arvanitis, Ch. Makridakis and A. E. Tzavaras, Stability and convergence of a class of finite element schemes for hyperbolic conservation laws, SIAM J. Numer. Anal. 42(4) (2004) 1357–1393]; [S. Jin and X. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277] for systems of hyperbolic conservation laws without source terms.

A Nonlinear Ultrasound Method for Fatigue Evaluation of Marine Structures

Fatigue is a major cause of failure in marine structures resulting from random wave and wind loading. A nonlinear ultrasound method for fatigue evaluation which uses interaction of two non-collinear nonlinear ultrasonic waves with quadratic nonlinearity is investigated in this paper. A hyperbolic system of conservation laws is applied here and a semi-discrete central scheme is used to solve the numerical problem. The numerical results prove that a resonant wave can be generated by two primary waves with certain resonant conditions. Features of the resonant wave are analyzed both in the time and frequency domains, and several regularities are found on intensity distribution of the resonant wave in two-dimensional domain.

Fast sweeping methods for hyperbolic systems of conservation laws at steady state II

The idea of using fast sweeping methods for solving stationary systems of conservation laws has previously been proposed for efficiently computing solutions with sharp shocks. We further develop these methods to allow for a more challenging class of problems including problems with sonic points, shocks originating in the interior of the domain, rarefaction waves, and two-dimensional systems. We show that fast sweeping methods can produce higher-order accuracy. Computational results validate the claims of accuracy, sharp shock curves, and optimal computational efficiency.

Convergence of two-dimensional staggered central schemes on unstructured triangular grids

In this paper, we present a convergence analysis of a two-dimensional central finite volume scheme on unstructured triangular grids for hyperbolic systems of conservation laws. More precisely, we show that the solution obtained by the numerical base scheme presents, under an appropriate CFL condition, an optimal convergence to the unique entropy solution of the Cauchy problem.

Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows

In this work, we present a fast and parallel finite volume scheme on unstructured meshes applied to complex fluid flow. The mathematical model is based on a three-dimensional compressible low Mach two-phase flows model, combined with a linearised ‘artificial pressure’ law. This hyperbolic system of conservation laws allows an explicit scheme, improved by a block-based adaptive mesh refinement scheme. Following a previous one-dimensional work, the useful numerical density of entropy production is used as mesh refinement criterion. Moreover, the computational time is preserved using a local time-stepping method. Finally, we show through several test cases the efficiency of the present scheme on two- and three-dimensional dam-break problems over an obstacle.

A Posteriori Analysis of Discontinuous Galerkin Schemes for Systems of Hyperbolic Conservation Laws

In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework. The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws. In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.

Entropy production and mesh refinement – application to wave breaking

We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. This approach is validated in the case of classical 1D numerical tests. Then, we present a 3D application. Indeed, according to an eulerian bi-fluid formulation at low Mach, an hyperbolic system of conservation laws allows an easily parallelization and mesh refinement by blocks .

Godunov scheme for Maxwell’s equations with Kerr nonlinearity

We study the Godunov scheme for a nonlinear Maxwell model arising in nonlinear optics, the Kerr model. This is a hyperbolic system of conservation laws with some eigenvalues of variable multiplicity, neither genuinely nonlinear nor linearly degenerate. The solution of the Riemann problem for the full-vector 6x6 system is constructed and proved to exist for all data. This solution is compared to the one of the reduced Transverse Magnetic model. The scheme is implemented in one and two space dimensions. The results are very close to the ones obtained with a Kerr-Debye relaxation approximation.

The high order control volume discontinuous Petrov–Galerkin finite element method for the hyperbolic conservation laws based on Lax–Wendroff time discretization

In this paper we constructed a high order control volume discontinuous finite element method for both scalar and systems of hyperbolic conservation laws based on Lax–Wendroff time discretization. The method combines advantages of both control volume discontinuous finite element methods and Lax–Wendroff time discretization. The method can preserve local conservation. It is high order and high resolution. The limiter is only used once in each temporal discretization step. Several numerical examples are used to demonstrate the accuracy and high resolution of the method.