Systems Of Nonlinear Conservation Laws Research Articles

Vanishing Viscosity with Short Wave–Long Wave Interactions for Systems of Conservation Laws

Motivated by Benney’s general theory, we propose new models for short wave–long wave interactions when the long waves are described by nonlinear systems of conservation laws. We prove the strong convergence of the solutions of the vanishing viscosity and short wave–long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems. We analyze several of the representative examples, such as scalar conservation laws, general symmetric systems, nonlinear elasticity and nonlinear electromagnetism.

Local Time Stepping Applied to Implicit-Explicit Methods for Hyperbolic Systems

In the context of systems of nonlinear conservation laws it can be crucial to use adaptive grids in order to correctly simulate the singularities of the solution over long time ranges while keeping the computing time within acceptable bounds. The adaptive space grid must vary in time according to the local smoothness of the solution. More sophisticated and recent methods also adapt the time-step locally to the space discretization according to the stability condition. We present here such a method designed for an explicit-implicit Lagrange-projection scheme, addressing physical problems where slow kinematic waves coexist with fast acoustic ones. Numerical simulations are presented to validate the algorithms in terms of robustness and efficiency.

Hyperbolic Conservation Laws

Hyperbolic Conservation Laws

Decay estimate for a new measure of rarefaction waves in general systems of conservation laws

The sharp decay estimate of rarefaction waves in terms of a partial ordering among positive measures is not only interesting in itself but also crucial in the study of convergence rate of vanishing viscosity approximations, cf. Bressan and Yang (2004) [10]. Such an estimate is well established for genuinely nonlinear system of conservation laws, cf. Bressan and Yang (2004) [9]. But similar result is not available for non-genuinely nonlinear system. In this paper, we give a new measure about the rarefaction waves. In addition, a sharp decay estimate of the new measure is given for the cubic nonlinear system of conservation laws.

Simple waves do not avoid eigenvalue crossings

AbstractGeneral one‐parameter families of matrices avoid eigenvalue crossings. It is shown that the matrices associated with the simple waves of nonlinear systems of conservation laws do not obey this rule: a subset of simple waves with nonzero measure has crossing eigenvalues. © 2009 Wiley Periodicals, Inc.

Optimal Control in Networks of Pipes and Canals

This paper deals with the optimal control of systems governed by nonlinear systems of conservation laws at junctions. The applications considered range from gas compressors in pipelines to open channels management. The existence of an optimal control is proved. From the analytical point of view, these results are based on the well posedness of a suitable initial boundary value problem and on techniques for quasidifferential equations in a metric space.

A kinematic model of continuous separation and classification of polydisperse suspensions

Kinematic models for polydisperse suspensions are based on specifying the solid–fluid relative velocity for each solids species as a function of the local solids concentrations. One such model, the Masliyah–Lockett–Bassoon (MLB) model, is employed herein to simulate continuous separation and classification of polydisperse suspensions. To this end, the clarifier-thickener (CT) setup for the continuous separation of suspensions is extended to a generalized clarifier-thickener (GCT). Discharge streams (or products) are described by new singular sink terms. Combining the GCT setup with the MLB model yields a system of nonlinear conservation laws with a discontinuous flux and a new non-conservative transport term describing the sinks. A numerical algorithm for the solution of this equation is presented along with numerical examples. The model describes the GCT unit with all critical design parameters, and predicts the composition of the overflow, underflow and discharge streams and the spatio-temporal evolution of the solids species concentrations inside the unit.

A Conservative Front Tracking Method in N-Dimensions

We propose a fully conservative Front Tracking algorithm for systems of nonlinear conservation laws. The algorithm can be applied uniformly in one, two, three and N dimensions. Implementation details for this algorithm and tests of fully conservative simulations are reported.

BOUNDARY CONTROL OF SYSTEMS OF CONSERVATION LAWS : LYAPUNOV STABILITY WITH INTEGRAL ACTIONS

A boundary control law with integral actions is proposed for a generic class of two-by-two homogeneous systems of linear conservation laws. Sufficient conditions on the tuning parameters are stated that guarantee the asymptotic stability of the closed-loop system. The closed-loop stability is analysed with an appropriate Lyapunov function. The control design method is validated with an experimental application to the regulation of water depth and flow rate in a pilot open-channel described by Saint-Venant equations. This hydraulic application shows that the control can be robustly implemented on nonhomogeneous systems of nonlinear conservation laws.

Distribution solutions of nonlinear systems of conservation laws

Distribution solutions of nonlinear systems of conservation laws

Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.

High-order discontinuous Galerkin methods using an hp-multigrid approach

The goal of this paper is to investigate and develop a fast and robust algorithm for the solution of high-order accurate discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured grids. Herein we present the development of a spectral hp-multigrid method, where the coarse “grid” levels are constructed by reducing the order ( p) of approximation of the discretization using hierarchical basis functions ( p-multigrid), together with the traditional ( h-multigrid) approach of constructing coarser grids with fewer elements. On each level we employ variants of the element-Jacobi scheme, where the Jacobian entries associated with each element are treated implicitly (i.e., inverted directly) and all other entries are treated explicitly. The methodology is developed for the two-dimensional non-linear Euler equations on unstructured grids, using both non-linear (FAS) and linear (CGC) multigrid schemes. Results are presented for the channel flow over a bump and a uniform flow over a four element airfoil. Current results demonstrate convergence rates which are independent of both order of accuracy ( p) of the discretization and level of mesh resolution ( h).

A variational multiscale finite element method for multiphase flow in porous media

We present a stabilized finite element method for the numerical solution of multiphase flow in porous media, based on a multiscale decomposition of pressures and fluid saturations into resolved (or grid) scales and unresolved (or subgrid) scales. The multiscale split is invoked in a variational setting, which leads to a rigorous definition of a grid scale problem and a subgrid scale problem. The subgrid problem is modeled using an algebraic approximation. This model requires the definition of a matrix of intrinsic time scales, which we design based on stability considerations. We illustrate the performance of the method with simulations of a waterflood in a heterogeneous oil reservoir. The proposed method yields stable, highly accurate solutions on very coarse grids, which we compare with those obtained by the classical Galerkin method or the upstream finite difference method. Although this paper is restricted to multiphase flow in porous media, the formulation is quite general and can be applied to other nonlinear systems of conservation laws, like the shallow water equations.

ADER schemes for three-dimensional non-linear hyperbolic systems

In this paper, we carry out the extension of the ADER approach to multidimensional non-linear systems of conservation laws. We implement non-linear schemes of up to fourth order of accuracy in both time and space. Numerical results for the compressible Euler equations illustrate the very high order of accuracy and non-oscillatory properties of the new schemes. Compared to the state-of-art finite-volume WENO schemes the ADER schemes are faster, more accurate, need less computer memory and have no theoretical accuracy barrier.

On the Numerical Simulation of Three‐Phase Reservoir Transport Problems

We introduce a new method for the numerical solution of three‐phase immiscible displacement in porous media. An operator splitting technique allows for the use of distinct time steps for the solution of advection and diffusion problems defined by the procedure. A nonlinear system of conservation laws is approximated by a nonoscillatory, second order, conservative central difference scheme in the advection step. Locally conservative mixed finite elements are used to discretize the nonlinear diffusion system. A simulator was developed to solve one‐dimensional problems and it was shown to be accurate and efficient through a set of numerical experiments. The numerical solutions are in very good agreement with semi‐analytic results available in the literature. In these numerical solutions transitional waves, which have a strong dependency upon the physical diffusion being modeled, have been captured.

On the convergence rate of vanishing viscosity approximations

AbstractGiven a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) −uϵ(t, ·)‖=O(1)(1 +t) ·$\sqrt{\varepsilon}$|ln ϵ| on the distance between an exact BV solutionuand a viscous approximationuϵ, letting the viscosity coefficient ϵ → 0. In the proof, starting fromuwe construct an approximation of the viscous solutionuϵby taking a mollificationu*$\varphi_{\sqrt{\varepsilon}}$and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ϵ. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.

Dynamic and steady-state sedimentation of polydisperse suspension and prediction of outlets particle-size distribution

The Kynch theorem was extended to non-linear system of conservation laws of polydisperse suspensions of spherical particles. The simulation predicts overflow of light particles and heavy particles at steady state and dynamic mode of batch and continuous sedimentation. The model eliminated the need for imposed non-theoretical parameters or functions to predict hindered settling and effluent concentrations. Particle-size distribution is also predicted at the effluent and the underflow. We examined several cases and predicted dynamic behaviour of rarefaction waves and overloaded continuous settler. A new concept of dynamic flux curves is also used and introduced.

A Sharp Decay Estimate for Positive Nonlinear Waves

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources.

A Posteriori Error Estimate for Front-Tracking: Systems of Conservation Laws

We demonstrate an a posteriori error estimate in the L1 norm for front-tracking approximate solutions to hyperbolic systems of nonlinear conservation laws. Starting with the L1 -stability result of Bressan, Liu, and Yang, we use their L1 -equivalent functional for pairs of front-tracking approximations and identify the leading order contribution to the numerical error. Our measure for the error explicitly identifies the local sources of errors within front-tracking approximations. We also show that these local error estimators are necessary and sufficient global error estimators. We apply the estimate to a new and wider class of front-tracking approximations, which includes the approximations of Risebro.

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L p and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.