In this paper we study the asymptotic nonlinear stability of discrete shocks for the Lax-Friedrichs scheme for approximating general m systems of nonlinear hyperbolic conservation laws. It is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the LP-norm for all p __> 1, provided that the sums of the initial perturbations equal zero. These results should shed light on the convergence of the numerical solution constructed by the Lax-Friedrichs scheme for the single-shock solution of system of hyperbolic conservation laws. If the Riemann solution corresponding to the given far-field states is a superposition of m single shocks from each characteristic family, we show that the corresponding multiple discrete shocks are nonlinearly stable in L p (p >= 2). These results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme.

/pdf/nonlinear-stability-of-discrete-shocks-for-systems-of-10jxl0hhpz.pdf

Nonlinear Stability of Discrete Shocks for Systems of Conservation Laws

Continuum shock profiles for discrete conservation laws I: Construction

The validity of the modified equation for nonlinear shock waves

The nonlinear stability of traveling Lax shocks in semidiscrete conservation laws involving general spatial forward-backward discretization schemes is considered. It is shown that spectrally stable semidiscrete Lax shocks are nonlinearly stable. In addition, it is proved that weak semidiscrete Lax profiles satisfy the spectral stability hypotheses made here and are therefore nonlinearly stable. The nonlinear stability results are proved by constructing the resolvent kernel using exponential dichotomies, which have recently been developed in this setting, and then using the contour integral representation for the associated Green's function to derive pointwise bounds that are sufficient for proving nonlinear stability. Previous stability analyses for semidiscrete shocks relied primarily on Evans functions, which exist only for one-sided upwind schemes.

/pdf/nonlinear-stability-of-semidiscrete-shocks-for-two-sided-5573dwa50p.pdf

Nonlinear Stability of Semidiscrete Shocks for Two-Sided Schemes

The nonlinear stability in the L[sup p]-norm, p [ge] 1, of stationary weak discrete shocks for the Lax-Friedrichs scheme approximating general m [times] m systems of nonlinear hyperbolic conservation laws is proved, provided that the summations of the initial perturbations equal zero. The result is proved by using a both weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the Lax-Friedrichs scheme. 13 refs.

/pdf/l-1-stability-of-stationary-discrete-shocks-58bj7yc4g4.pdf

L^1 stability of stationary discrete shocks

Discrete shocks for difference approximations to systems of conservation laws

Strong viscous shocks for systems of conservation laws with a high order of dissipation

The Kuramoto-Sivashinsky equation 

$$ {u_t} + {u_{XXXX}} + {u_{XX}} + _2^1u_X^2 = 0, - \infty 0 $$

(1.1)

has attracted for the last decade a considerable attention ([3]–[6], [8], [12], [14]–[17]). It was originally derived by Kuramoto and Tsuzuki [3] in the context of a reaction diffusion system, and by Sivashinsky [14] in the context of flame front propagation. In the latter case u(x,t) represents the perturbation of a flame front which propagates in a fuel — oxygen mixture. Numerical experiments ([4], [6], [8], [17]) have shown that eg. (1.1) when solved on a sufficiently large interval — l < X < l with periodic boundary conditions tends to a turbulent state as t → ∞. The solution u(x,t) has the form 

$$ u\left( {x.t} \right) = - c_0^2t + (x,t) $$

(1.2)

Where $ c_0^2 \approx 1.2 $ is a universal constant independent of the initial condition, while the mean value of v(x,t) is close to zero. For a fixed t the function v(x,t), although irregular, has an appearance of a quasi-periodic wave with a characteristic wave length $ {\ell _0} = 2\pi /{\omega _0},{\omega _0} = \sqrt 2 /2 $ Note that the frequency ω0 is maximally amplified by the linear terms in (1.1). Formula (1.2) suggests that one should look for steady solutions of (1.1), namely solutions of the form 

$$ u(x,t) = - {c^2}t + v(x) $$

(1.3)

Clearly v(x) satisfies the O.D.E. 

$$ \frac{{{d^4}v}}{{d{x^4}}} + \frac{{{d^2}v}}{{d{x^2}}} = {c^2} - \frac{1}{2}{(\frac{{dc}}{{dx}})^2} $$

(1.4)

or a third order equation for the slope $ y = \frac{{dv}}{{dx}} $

$$ \frac{{{d^3}y}}{{d{x^3}}} + \frac{{dy}}{{dx}} = {c^2} - \frac{1}{2}{y^2}, - \infty < x < \infty $$

(1.5)

The main objective of this work is to study the set of bounded solutions of (1.5) and its dependence on the parameter c.

Order and Disorder in the Kuramoto-Sivashinsky Equation

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Papers

Discrete shocks for difference approximations to systems of conservation laws

Strong viscous shocks for systems of conservation laws with a high order of dissipation

Order and Disorder in the Kuramoto-Sivashinsky Equation