Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes

Journal of Differential Equations

2009

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Critical threshold phenomena in one-dimensional 2 × 2 quasi-linear hyperbolic relaxation systems are investigated. Assuming both the subcharacteristic condition and genuine nonlinearity of the flux, we prove global in time regularity and finite-time singularity formation of solutions simultaneously by showing the critical threshold phenomena associated with the underlying relaxation systems. Our results apply to the well-known isentropic Euler system with damping. Within the same framework it is also shown that the solution of the semi-linear relaxation system remains smooth for all time, provided the subcharacteristic condition is satisfied.

Section: Assumption 1 For All V Under Consideration It Holdsmentioning

confidence: 98%

Critical thresholds in hyperbolic relaxation systems

Journal of Differential Equations

2009

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“…Hyperbolic relaxation systems belong to a special class of balance laws, for which a sub-characteristic type condition is always necessary for even linear stability [43]. An abundant research on nonlinear stability theory for various relaxation systems has appeared in past decades, see e.g., [1,3,14,17,28,27,22,23,35,38], relying on some sub-characteristic type structure conditions [35].…”

Section: Remarkmentioning

confidence: 99%

Critical thresholds in a relaxation system with resonance of characteristic speeds

Discrete &Amp; Continuous Dynamical Systems - A

2009

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We study critical threshold phenomena in a dynamic continuum traffic flow model known as the Payne and Whitham (PW) model. This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system. For a scenario of physical interest we identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.

“…In particular, convergence and stability of relaxation schemes introduced in [12] have been well justified, see, e.g. [17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

An Alternating Evolution Approximation to Systems of Hyperbolic Conservation Laws

J. Hyper. Differential Equations

2008

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In this paper, we present an alternating evolution (AE) approximationto systems of hyperbolic conservation lawsin arbitrary spatial dimension. We prove the convergence of the approximate solutions towards an entropy solution of scalar multi-D conservation laws, and the L 1 contraction property for the approximate solution is established as well. It is also shown that such an approximation is extremely accurate in the sense that if initial data is prepared such that u 0 = v 0 = U 0 , then no method error is induced as time evolves, and the exact entropy solution is precisely captured. Furthermore, in the approximation system time evolution of one variable is associated with spatial redistribution in another variable. These features render such an approximation ideal to be used for construction of high resolution numerical schemes to solve hyperbolic conservation laws. The usual obstacles caused by jumps crossing computational cell interfaces are not felt when both u and v are sampled alternatively, and reconstructed independently. Herewith we discuss the designing principle for constructing AE schemes, with illustration of two preliminary schemes for systems of conservation laws in one dimension. Both l ∞ monotonicity and the TVD (Total Variational Diminishing) property are established for these schemes when applied to the scalar laws.