Strong instability of solitary waves for nonlinear Klein–Gordon equations and generalized Boussinesq equations

We study here instability problems of standing waves for the nonlinear Klein–Gordon equations and solitary waves for the generalized Boussinesq equations. It is shown that those special wave solutions may be strongly unstable by blowup in finite time, depending on the range of the wave's frequency or the wave's speed of propagation and on the nonlinearity.     

Higher order shadow waves and delta shock blow up in the Chaplygin gas

The introductory part of this paper contains an overview of known results about elementary and delta shock solutions to Riemann problem for well known Chaplygin gas model (nowadays used in cosmological theories for dark energy) in terms of entropic shadow waves. Shadow waves are introduced in [17] and they are represented by shocks depending on a small parameter ε with unbounded amplitudes having a distributional limit involving the Dirac delta function. In a search for admissible solutions to all possible cases of mutual interactions of waves arising from double Riemann initial data we found same cases that cannot be resolved with already known types of elementary or shadow wave solutions. These cases are resolved by introducing a sequence of higher order shadow waves depending on integer powers of ε. It is shown that such waves have a distributional limit but only until some finite time T.     

Nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation

In was shown in Ruan et al. (2008) [3] that rarefaction waves for the generalized KdV-Burgers-Kuramoto equation are nonlinearly stable provided that both the strength of the rarefaction waves and the initial perturbation are sufficiently small. The main purpose of this work is concerned with nonlinear stability of strong rarefaction waves for the generalized KdV-Burgers-Kuramoto equation with large initial perturbation. In our results, we do not require the strength of the rarefaction waves to be small and when the smooth nonlinear flux function satisfies certain growth condition at infinity, the initial perturbation can be chosen arbitrarily in , while for a general smooth nonlinear flux function, we need to ask for the L²-norm of the initial perturbation to be small but the L²-norm of the first derivative of the initial perturbation can be large and, consequently, the -norm of the initial perturbation can also be large.     

Interactions of nonlinear ion-acoustic waves in a collisionless plasma

In the present work, based on a one-dimensional model, the interaction of two solitary waves propagating in opposite directions in a collisionless plasma is investigated by use of the extended Poincaré–Lighthill–Kuo (PLK) method. It is shown that bi-directional solitary waves are propagated and the head-on collision of these two solitons occur. The phase shifts and the trajectories of these two solitons after the collision are obtained.     

Stability of the Camassa-Holm solitons

Summary. We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances.     

Local well-posedness and stability of peakons for a generalized Dullin-Gottwald-Holm equation

We establish the local well-posedness for a generalized Dullin-Gottwald-Holm equation by using Kato’s theory. Furthermore, the orbital stability of the peaked solitary waves is also proved.     

Interaction of rarefaction waves in jet stream

In this paper we present a mathematical analysis of a supersonic jet stream out of an orifice into the atmosphere. The analysis involves the interaction of steady rarefaction waves, and the interaction of a rarefaction wave by the interface of the jet stream. The existence of the classical solution in the region of interactions of rarefaction waves is established. For small pressure difference the existence of the classical solution in the region of reflection is also obtained. Finally, for large pressure difference vacuum may be produced by strong expanding, and the corresponding wave structure with vacuum is also analyzed.     

A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation

In this article, a reduced optimizing finite difference scheme (FDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for Burgers equation is presented. Also the error estimates between the usual finite difference solution and the POD solution of reduced optimizing FDS are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between the POD solution of reduced optimizing FDS and the solution of the usual FDS is consistent with theoretical results. Moreover, it is also shown that the reduced optimizing FDS is feasible and efficient.     

Global structure and asymptotic behavior of weak solutions to flood wave equations

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Solitary waves and a stability analysis of an equation of short and long dispersive waves

A universal model for the interaction of long nonlinear waves and packets of short waves with long linear carrier waves is given by a system in which an equation of Korteweg–de Vries (KdV) type is coupled to an equation of nonlinear Schrödinger (NLS) type. The system has solutions of steady form in which one component is like a solitary-wave solution of the KdV equation and the other component is like a ground-state solution of the NLS equation. We study the stability of solitary-wave solutions to an equation of short and long waves by using variational methods based on the use of energy–momentum functionals and the techniques of convexity type. We use the concentration compactness method to prove the existence of solitary waves. We prove that the stability of solitary waves is determined by the convexity or concavity of a function of the wave speed.     

Conservation law with the flux function discontinuous in the space variable—II: Convex–concave type fluxes and generalized entropy solutions

We deal with a single conservation law with discontinuous convex–concave type fluxes which arise while considering sign changing flux coefficients. The main difficulty is that a weak solution may not exist as the Rankine–Hugoniot condition at the interface may not be satisfied for certain choice of the initial data. We develop the concept of generalized entropy solutions for such equations by replacing the Rankine–Hugoniot condition by a generalized Rankine–Hugoniot condition. The uniqueness of solutions is shown by proving that the generalized entropy solutions form a contractive semi-group in L¹$L^1$. Existence follows by showing that a Godunov type finite difference scheme converges to the generalized entropy solution. The scheme is based on solutions of the associated Riemann problem and is neither consistent nor conservative. The analysis developed here enables to treat the cases of fluxes having at most one extrema in the domain of definition completely. Numerical results reporting the performance of the scheme are presented.     

Numerical pricing of options using high-order compact finite difference schemes

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.     

A Multi-Symplectic Scheme for RLW Equation

The Hamiltonian and multi-symplectic formulations for RLW equation are considered in this paper. A new twelve-point difference scheme which is equivalent to multi-symplectic Preissmann integrator is derived based on the multi-symplectic formulation of RLW equation. And the numerical experiments on solitary waves are also given. Comparing the numerical results for RLW equation with those for KdV equation, the inelastic behavior of RLW equation is shown.     

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.     

A fully discrete scheme for diffusive-dispersive conservation laws

Summary.   We introduce a fully discrete (in both space and time) scheme for the numerical approximation of diffusive-dispersive hyperbolic conservation laws in one-space dimension. This scheme extends an approach by LeFloch and Rohde [4]: it satisfies a cell entropy inequality and, as a consequence, the space integral of the entropy is a decreasing function of time. This is an important stability property, shared by the continuous model as well. Following Hayes and LeFloch [2], we show that the limiting solutions generated by the scheme need not coincide with the classical Oleinik-Kruzkov entropy solutions, but contain nonclassical undercompressive shock waves. Investigating the properties of the scheme, we stress various similarities and differences between the continuous model and the discrete scheme (dynamics of nonclassical shocks, nucleation, etc). Received November 15, 1999 / Revised version received May 27, 2000 / Published online March 20, 2001     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Finite element simulations of window Josephson junctions

This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed.     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.