Zero Diffusion-Dispersion Limits for Scalar Conservation Laws

Zero Diffusion-dispersion-smoothing Limits for Scalar Conservation Law with Discontinuous Flux Function

We consider multi-dimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of Hmeasures [17] to investigate the zero diffusion-dispersion-smoothing limit. Content

Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

Moderate dispersion in scalar conservation laws

We consider the weakly dissipative and weakly dispersive Burgers-Hopf-Korteweg-de-Vries equation with the diffusion coefficient ε and the dispersion rate δ in the range δ/ε → 0. We study the travelling wave connecting u(−∞) = 1 to u(+∞) = 0 and show that it converges strongly to the entropic shock profile as ε, δ → 0. Key-words Travelling waves, moderate dispersion, Korteweg de Vries equation, ...

A Singular Limit Problem for Conservation Laws Related to the Camassa-holm Shallow Water Equation

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove that as the diffusion and dispersion parameters tend to zero, with a condition on the relative balance between these two parameters, smooth solutions of the shallow water equation converge to discontinuous solutions of a scalar conservation law...

A total variation diminishing high resolution scheme for nonlinear conservation laws

In this paper we propose a novel high resolution scheme for scalar nonlinear hyperbolic conservation laws. The aim of high resolution schemes is to provide at least second order accuracy in smooth regions and produce sharp solutions near the discontinuities. We prove that the proposed scheme that is derived by utilizing an appropriate flux limiter is nonlinear stable in the sense of total varia...