Relaxation Schemes For Conservation Laws Research Articles

Numerical study of phase transition in van der Waals fluid

In this article, we use a relaxation scheme for conservation laws to study liquid-vapor phase transition modeled by the van der Waals equation, which introduces a small parameter $e$ and a new variable. We solve the relaxation system in Lagrangian coordinates for one dimension and solve the system in Eulerian coordinates for two dimension. A second order TVD Runge-Kutta splitting scheme is used in time discretization and upwind or MUSCL scheme is used in space discretization. The long time behavior of the fluid is numerically investigated. If the initial data belongs to elliptic region, the solution converges to two Maxwell states. When the initial data lies in metastable region, the solution either remains in the same phase or converges to the Maxwell states depending to the initial perturbation. If the initial state is in the stable region, the solution remains in that region for all time.

A relaxation scheme for conservation laws with a discontinuous coefficient

lWe study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k E BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat-Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 x 2 Riemann solver.

Convergence analysis of relaxation schemes for conservation laws with stiff source terms

We analyze the convergence for relaxation approximation applied to conservation laws with sti source terms. We suppose that the source term q(u) is dissipative. Semi-implicit relaxing schemes are investigated and the corresponding stability theory is established. In particular, we proved that the numerical solution of a rst-order relaxing scheme is uniformlly l 1 , l 1 and TV- stable, in the sense that they can be bounded by a constant independent of the the relaxation parameter and the the Lipschitz constant of the sti source term, and time step t. Concergence of the relaxing scheme is then established. The results obtained for the rst-order relaxing scheme can be extended to MUSCL relaxing schemes.

Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations

Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diffusive scaling that leads to the diffusion equations. In many physical applications, the scaling parameter (mean free path) may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes within one problem, and it is desirable to develop a class of robust numerical schemes that can work uniformly with respect to this relaxation parameter. In an earlier work [Jin, Pareschi, and Toscani, SIAM J. Numer. Anal., 35 (1998), pp. 2405--2439] we handled this numerical problem for discrete-velocity kinetic models by reformulating the system into a form commonly used for a relaxation scheme for conservation laws [Jin and Xin, Comm. Pure Appl. Math., 48 (1995), pp. 235--277]. Such a reformulation allows us to use the splitting technique for relaxation schemes to design a class of implicit, yet explicitly implementable, schemes that work with high resolution uniformly with respect to the relaxation parameter. In this paper we show that such a numerical technique can be applied to a large class of transport equations with continuous velocities, when one uses the even and odd parities of the transport equation.

Two‐layer Gravity Currents with Topography

Two‐dimensional and time‐dependent gravity currents involving the initial release of a fixed volume of heavy fluid over a gradually sloping bottom and underlying a layer of lighter fluid are considered. The equations which describe the resulting two‐layer flow are derived from the Navier–Stokes equations for a constant density, inviscid, nonrotating fluid, neglecting kinematic viscosity, surface tension, and entrainment between the layers. A new addition to the theory is introduced in the form of a forcing term in the lower layer horizontal momentum equation which is incorporated to produce the characteristic structure typical of such gravity currents in the laboratory. This delaying term is restricted to the front of the gravity current, and as such is shown to be valid under conventional shallow‐water scaling assumptions. The hyperbolic character of the equations of motion is shown, a simple numerical test for hyperbolicity is derived from theoretical considerations, and these results are related to the stability Froude number of the flow. Well‐posedness of the initial boundary value problem is proven via localization of the equations, and the discussion is extended to a two‐point boundary value problem with examples of steady‐state and traveling wave solutions given for a bottom surface of constant slope. Numerical results are obtained by using a recently developed finite‐difference relaxation scheme for conservation laws, sufficiently modified herein to include spatial variability and forcing terms, which approximates the material interface at the front of the lower fluid layer as a shock. The effects of slope and the delaying force are investigated numerically to determine their theoretical importance, and the range of expected values is compared to published experimental results. Some calculations for the temporal evolution of the flow are produced that display the phenomenon of rear wall separation for nonzero slopes.

Analytical and numerical results for flow and shock formation in two-layer gravity currents

AbstractMany gravity driven flows can be modelled as homogeneous layers of inviscid fluid with a hydrostatic pressure distribution. There are examples throughout oceanography, meteorology, and many engineering applications, yet there are areas which require further investigation. Analytical and numerical results for two-layer shallow-water formulations of time dependent gravity currents travelling in one spatial dimension are presented. Model equations for three physical limits are developed from the hydraulic equations, and numerical solutions are produced using a relaxation scheme for conservation laws developed recently by S. Jin and X. Zin [6]. Hyperbolicity of the model equations is examined in conjunction with the stability Froude number, and shock formation at the interface of the two layers is investigated using the theory of weakly nonlinear hyperbolic waves.

Convergence to equilibrium for the relaxation approximations of conservation laws

We study the Cauchy problem for 2 × 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0. This research was strongly motivated by the recent numerical investigations of S. Jin and Z. Xin on the relaxation schemes for conservation laws. © 1996 John Wiley & Sons, Inc.

Convergence of relaxation schemes for conservation laws

We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L∞ and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated.