Nonlinear Diffusion Wave Research Articles

Pointwise estimate of solutions of isentropic Navier-Stokes equations in even space-dimensions

This paper is concerned with the dissipation of solutions of the isentropic Navier-Stokes equations in even and bigger than two multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and the generalized Huygan's principle is exhibited. The approch of the paper is based on the detailed analysis of the Green function of linearized system. This is used to study the coupling of nonlinear diffusion waves.

Nonlinear Diffusive Wave Modeling and Identification of Open Channels

This paper deals with the identification of an irrigation system in the presence of not completely known withdrawals. The modeling approach is based on a time-implicit finite-dimensional model of simplified equations governing open-channel hydraulics. The diffusive wave model is used with a parameterization based on large rectangular geometry and solved using the Muskingum-Cunge method. A two-stage iterative method is proposed to identify the dynamic parameters and the withdrawal parameters. In the first stage, only the parameters governing the withdrawals are estimated, and in the second stage, the estimated withdrawals are used to estimate the parameters of the dynamics. The simulation studies performed on simulated and real data show the effectiveness of the identification method.

Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping

In this paper we consider a model of hyperbolic balance laws with damping on the quarter plane ( x , t ) ∈ R + × R + (x, t) \in {\mathbb {R}_ + } \times {\mathbb {R}_ + } . By means of a suitable shift function, which will play a key role to overcome the difficulty of large boundary perturbations, we show that the IBVP solutions converge time-asymptotically to the shifted nonlinear diffusion wave solutions of the Cauchy problem to the nonlinear parabolic equation given by the related Darcy’s law. We obtain also the time decay rates, which are the optimal ones in the L 2 {L^{2}} -sense. Our proof is based on the use of the classical energy method.

Lp-Convergence Rate to Nonlinear Diffusion Waves for p-System with Damping

In this paper, we study the p-system with frictional damping and show that the solutions time-asymptotically tend to the nonlinear diffusion waves governed by the classical Darcy's law. By introducing an approximate Green function, we obtain the optimal Lp, 2⩽p⩽+∞, convergence rate of the solution, which is a perturbation of the nonlinear diffusion wave, to the hyperbolic system.

The Pointwise Estimates of Diffusion Wave for the Navier-Stokes Systems in Odd Multi-Dimensions

We study the dissipation of solutions of the isentropic Navier–Stokes equations in odd multi-dimensions. Pointwise estimates of the time-asymptotic shape of the solutions are obtained and shown to exhibit the generalized Huygen's principle. Our approach is based on the detailed analysis of the Green function of the linearized system. This is used to study the coupling of nonlinear diffusion waves.

Pointwise convergence to shock waves for viscous conservation laws

We are interested in the pointwise behavior of the perturbations of shock waves for viscous conservation laws. It is shown that, besides a translation of the shock waves and of linear and nonlinear diffusion waves of heat and Burgers equations, a perturbation also gives rise to algebraically decaying terms, which measure the coupling of waves of different characteristic families. Our technique is a combination of time-asymptotic expansion, construction of approximate Green functions, and analysis of nonlinear wave interactions. The pointwise estimates yield optimal Lp convergence of the perturbation to the shock and diffusion waves, 1 ≤ p ≤ ∞. The new approach of obtaining pointwise estimates based on the Green functions for the linearized system and the analysis of nonlinear wave interactions is also useful for studying the stability of waves of distinct types and nonclassical shocks. These are being explored elsewhere. © 1997 John Wiley & Sons, Inc.

Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt → 0, the space step size Δx = Q/(2S 0 BC k ) where Q is discharge, S 0 is bed slope, B is channel width and C k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs.

Accurate Diffusive Wave Routing

The diffusive wave simplification of the unsteady, open-channel flow equations is a commonly used approach for flood routing applications. Among the many routing methods, except for those based on the De Saint-Venant equations, it may be considered to be the one that most closely complies with the physics of open-channel hydraulics. However, common implementations of the diffusive wave approach (linear diffusion, variable parameter diffusion) involve additional approximations that diminish its physical significance and accuracy. A new formulation for the nonlinear diffusive wave is presented that better respects conservation principles through close consistency with the fundamental flow equations. The accuracy and reliability of the proposed model are shown on test cases consisting both of a hypothetical, regular channel, and of an actual river reach. Simulated discharges are compared to those obtained by a full De Saint-Venant model and by ordinary diffusion methods, as well as to observed hydrographs in...

Convergence Rates to Nonlinear Diffusion Waves for Solutions of System of Hyperbolic Conservation Laws with Damping

Convergence Rates to Nonlinear Diffusion Waves for Solutions of System of Hyperbolic Conservation Laws with Damping

Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping

We consider a model of hyperbolic conservation laws with damping and show that the solutions tend to those of a nonlinear parabolic equation time-asymptotically. The hyperbolic model may be viewed as isentropic Euler equations with friction term added to the momentum equation to model gas flow through a porous media. In this case our result justifies Darcy's law time-asymptotically. Our model may also be viewed as an elastic model with damping.

Large-time behaviour of solutions to hyperbolic–parabolic systems of conservation laws and applications

SynopsisWe study the large-time behaviour of solutions to the initial value problem for hyperbolic-parabolic systems of conservation equations in one space dimension. It is proved that under suitable assumptions a unique solution exists for all time t ≧ 0, and converges to a given constant state at the rate t − ¼ as t → ∞. Moreover, it is proved that the solution approaches the superposition of the non-linear and linear diffusion waves constructed in terms of the self-similar solutions to the Burgers equation and the linear heat equation at the rate t − ½ +α, α > 0, as t →∞. The proof is essentially based on the fact that for t → ∞ the solution to the hyperbolic-parabolic system is well approximated by the solution to a semilinear uniformly parabolic system whose viscosity matrix is uniquely determined from the original system. The results obtained are applicable straightforwardly to the equations of viscous (or inviscid) heat-conductive fluids.

Shock waves for compressible navier‐stokes equations are stable

AbstractIt is shown that shock waves for the compressible Navier‐Stokes equations are nonlinearly stable. A perturbation of a shock wave tends to the shock wave, properly translated in phase, as time tends to infinity. Through the consideration of conservation of mass, momentum and energy we obtain an a priori estimate of the amount of translation of the shock wave and the strength of the linear and nonlinear diffusion waves that arise due to the perturbation. Our techniques include the energy method for parabolic‐hyperbolic systems, the decomposition of waves, and the energy‐characteristic method for viscous conservation laws introduced earlier by the author.