Transport‐equilibrium schemes for computing nonclassical shocks. Scalar conservation laws

This paper presents a new numerical strategy for computing the nonclassical weak solutions of scalar conservation laws which fail to be genuinely nonlinear. We concentrate on the typical situation of concave–convex and convex–concave flux functions. In such situations the so‐called nonclassical shocks, violating the classical Oleinik entropy criterion and selected by a prescribed kinetic relation, naturally arise in the resolution of the Riemann problem. Enforcing the kinetic relation from a numerical point of view is known to be a crucial but challenging issue. By means of an algorithm made of two steps, namely an Equilibrium step and a Transport step, we show how to force the validity of the kinetic relation at the discrete level. The proposed strategy is based on the use of a numerical flux function and random numbers. We prove that the resulting scheme enjoys important consistency properties. Numerous numerical evidences illustrate the validity of our approach. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Enhanced-accuracy difference schemes for one nonclassical boundary-value problem

Translated from Aktual'nye Voprosy Prikladnoi Matematiki, pp. 93–97, 1989.     

On the construction of nonclassical finite-difference schemes for ordinary differential equations

We consider the Cauchy problem for stiff systems of ordinary differential equations. An essentially new method is suggested for constructing finite-difference schemes for such problems. These methods have a number of advantages over earlier developed schemes.     

Hybrid schemes with high-order multioperators for computing discontinuous solutions

Results are presented concerning high-order multioperator schemes and their monotonized versions as applied to the computation of discontinuous solutions. Two types of hybrid schemes are considered. Solutions of several test problems, including those with extremely strong discontinuities, are presented. An example of solving the Navier-Stokes equations at low supersonic Mach numbers by applying multioperator schemes without monotonization is given.     

The global decay to discrete shocks for scalar monotone schemes

Given a family of discrete shocks of a monotone scheme, we prove that the discrete shock profile with rational shock speed is asymptotically stable with respect to the topology : if , then as under no restriction conditions of the initial data to the interval . The asymptotic wave profile is uniquely identified from the above family by a mass parameter.

     

Fourth-order compact schemes for a parabolic-ordinary system of European option pricing liquidity shocks model

This paper provides a numerical investigation for European options under parabolic-ordinary system modeling markets to liquidity shocks. Our main results concern construction and analysis of fourth order in space compact finite difference schemes (CFDS). Numerical experiments using Richardson extrapolation in time are discussed.     

Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces

Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak^? continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge ?. Let f be an α-contraction and {Tn} a sequence of nonexpansive mappings, we study the strong convergence of explicit iterative schemes

(1)     

The use of high-order composite compact schemes for computing supersonic jet interaction with a surface

High-order composite compact schemes are applied to the simulation of viscous gas dynamics with strong discontinuities of flow variables. To perform shock-capturing computations of such problems, the dissipation of the basic operators is enhanced and the solutions obtained with these operators are locally replaced by those produced with the help of simple one-sided differences. Numerical results obtained for the shock interaction of a supersonic axisymmetric jet with a flat surface are presented.     

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.     

Boundary-value problems for some higher-order nonclassical differential equations

The paper consists of two parts. The first part deals with the solvability of new boundary-value problems for the model quasihyperbolic equations (?1)^p D _t ^2p u = Au + f(x, t), where p > 1, for a self-adjoint second-order elliptic operator A. For the problems under study, the existence and uniqueness theorems are proved for regular solutions. In the second part, the results obtained in the first part are somewhat sharpened and generalized.     

Classical and nonclassical symmetries for the Krichever-Novikov equation

We study the Krichever-Novikov equation from the standpoint of the theory of symmetry reductions in partial differential equations. We obtain a Lie group classification. Moreover, we obtain some exact solutions, and we apply the nonclassical method.     

Trajectory attractors for nonclassical diffusion equations with fading memory

In this article, we consider the existence of trajectory and global attractors for nonclassical diffusion equations with linear fading memory. For this purpose, we will apply the method presented by Chepyzhov and Miranville [7,8], in which the authors provide some new ideas in describing the trajectory attractors for evolution equations with memory.     

Attractors for the nonclassical diffusion equations with fading memory

We discuss long-time dynamical behavior of the nonclassical diffusion equation with fading memory when nonlinearity is critical. The existence and regularity of global attractors in weak topological space and strong topological space are obtained, while the forcing term only belongs to H⁻¹(Ω) and L²(Ω) respectively. The results in this part are new and appear to be optimal corresponding to the forcing term.