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Register a new userThe Replacement Rule for Nonlinear Shallow Water Waves
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When a $(1+1)$-dimensional nonlinear PDE in real function $eta(x,t)$ admits localized traveling solutions we can consider $L$ to be the average width of the envelope, $A$ the average value of the amplitude of the envelope, and $V$ the group velocity of such a solution. The replacement rule (RR or nonlinear dispersion relation) procedure is able to provide a simple qualitative relation between these three parameters, without actually solve the equation. Examples are provided from KdV, C-H and BBM equations, but the procedure appears to be almost universally valid for such $(1+1)$-dimensional nonlinear PDE and their localized traveling solutions cite{3}.
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Surface and interfacial weakly-nonlinear ring waves in a two-layer fluid are modelled numerically, within the framework of the recently derived 2+1-dimensional cKdV-type equation. In a case study, we consider concentric waves from a localised initial condition and waves in a 2D version of the dam-break problem, as well as discussing the effect of a piecewise-constant shear flow. The modelling shows, in particular, the formation of 2D dispersive shock waves (DSWs) and oscillatory wave trains. The surface and interfacial DSWs generated in our numerical experiments look distinctively different.
We theoretically describe the quasi one-dimensional transverse spreading of a light pulse propagating in a nonlinear optical material in the presence of a uniform background light intensity. For short propagation distances the pulse can be described within a nondispersive approximation by means of Riemanns approach. For larger distances, wave breaking occurs, leading to the formation of dispersive shocks at both ends of the pulse. We describe this phenomenon within Whitham modulation theory, which yields an excellent agreement with numerical simulations. Our analytic approach makes it possible to extract the leading asymptotic behavior of the parameters of the shock.
We consider evolution of wave pulses with formation of dispersive shock waves in framework of fully nonlinear shallow-water equations. Situations of initial elevations or initial dips on the water surface are treated and motion of the dispersive shock edges is studied within the Whitham theory of modulations. Simple analytical formulas are obtained for asymptotic stage of evolution of initially localized pulses. Analytical results are confirmed by exact numerical solutions of the fully nonlinear shallow-water equations.
In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.
We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.
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