Schur rings and non-symmetric association schemes on 64 vertices

In this paper we enumerate essentially all non-symmetric association schemes with three classes, less than 96 vertices and with a regular group of automorphisms. The enumeration is based on a computer search in Schur rings. The most interesting cases have 64 vertices.In one primitive case and in one imprimitive case where no association scheme was previously known we find several new association schemes. In one other imprimitive case with 64 vertices we find association schemes with an automorphism group of rank 4, which was previously assumed not to be possible.     

A composite semi-conservative scheme for hyperbolic conservation laws

In this work a first order accurate semi-conservative composite scheme is presented for hyperbolic conservation laws. The idea is to consider the non-conservative form of conservation law and utilize the explicit wave propagation direction to construct semi-conservative upwind scheme. This method captures the shock waves exactly with less numerical dissipation but generates unphysical rarefaction shocks in case of expansion waves with sonic points. It shows less dissipative nature of constructed scheme. In order to overcome it, we use the strategy of composite schemes. A very simple criteria based on wave speed direction is given to decide the iterations. The proposed method is applied to a variety of test problems and numerical results show accurate shock capturing and higher resolution for rarefaction fan.     

Bounds for visual cryptography schemes

In this paper, we investigate the best pixel expansion of various models of visual cryptography schemes. In this regard, we consider visual cryptography schemes introduced by Tzeng and Hu (2002) [13]. In such a model, only minimal qualified sets can recover the secret image and the recovered secret image can be darker or lighter than the background. Blundo et al. (2006) [4] introduced a lower bound for the best pixel expansion of this scheme in terms of minimal qualified sets. We present another lower bound for the best pixel expansion of the scheme. As a corollary, we introduce a lower bound, based on an induced matching of hypergraph of qualified sets, for the best pixel expansion of the aforementioned model and the traditional model of visual cryptography scheme realized by basis matrices. Finally, we study access structures based on graphs and we present an upper bound for the smallest pixel expansion in terms of strong chromatic index.     

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic (0, p). As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic (0, p) of integral canonical models of projective Shimura varieties of Hodge type with respect to h-hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.     

Sustainability of pension systems with voluntary participation

Motivated by declining support for mandatory participation in pension arrangements, we explore whether the intergenerational risk-sharing benefits that these arrangements offer suffice to ensure their survival when participation becomes voluntary. Funded systems with asset buffers are particularly interesting since these buffers make contributions more sensitive to financial returns. Equilibria are characterised by thresholds on the young’s willingness to contribute. Standard values for our parameters yield two such equilibria; only the one with the higher threshold is consistent with the initial young being prepared to start the system. An advancement relative to the related literature is that the equilibria feature a non-zero probability of collapse. Finally, we explore the social welfare maximising values for the pension parameters for various levels of uncertainty and risk aversion.     

Unconditionally maximum principle preserving finite element schemes for the surface Allen–Cahn type equations

In this paper, we present two types of unconditionally maximum principle preserving finite element schemes to the standard and conservative surface Allen–Cahn equations. The surface finite element method is applied to the spatial discretization. For the temporal discretization of the standard Allen–Cahn equation, the stabilized semi-implicit and the convex splitting schemes are modified as lumped mass forms which enable schemes to preserve the discrete maximum principle. Based on the above schemes, an operator splitting approach is utilized to solve the conservative Allen–Cahn equation. The proofs of the unconditionally discrete maximum principle preservations of the proposed schemes are provided both for semi- (in time) and fully discrete cases. Numerical examples including simulations of the phase separations and mean curvature flows on various surfaces are presented to illustrate the validity of the proposed schemes.     

A nonuniform L2 formula of Caputo derivative and its application to a fractional Benjamin–Bona–Mahony-type equation with nonsmooth solutions

To recover the full accuracy of discretized fractional derivatives, nonuniform mesh technique is a natural and simple approach to efficiently resolve the initial singularities that always appear in the solutions of time-fractional linear and nonlinear differential equations. We first construct a nonuniform L2 approximation for the fractional Caputo's derivative of order 1 < α < 2 and present a global consistency analysis under some reasonable regularity assumptions. The temporal nonuniform L2 formula is then utilized to develop a linearized difference scheme for a time-fractional Benjamin–Bona–Mahony-type equation. The unconditional convergence of our scheme on both uniform and nonuniform (graded) time meshes are proven with respect to the discrete H¹-norm. Numerical examples are provided to justify the accuracy.     

The numerical analysis of two linearized difference schemes for the Benjamin–Bona–Mahony–Burgers equation

In the article, two linearized finite difference schemes are proposed and analyzed for the Benjamin–Bona–Mahony–Burgers (BBMB) equation. For the construction of the two-level scheme, the nonlinear term is linearized via averaging k and k + 1 floor, we prove unique solvability and convergence of numerical solutions in detail with the convergence order O(τ² + h²) . For the three-level linearized scheme, the extrapolation technique is utilized to linearize the nonlinear term based on ψ function. We obtain the conservation, boundedness, unique solvability and convergence of numerical solutions with the convergence order O(τ² + h²) at length. Furthermore, extending our work to the BBMB equation with the nonlinear source term is considered and a Newton linearized method is inserted to deal with it. The applicability and accuracy of both schemes are demonstrated by numerical experiments.     

Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.     

A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations

Among the typical time integrations for PDEs, Leap-frog scheme is the well-known method which can easily be used. A most welcome feature of the Leap-frog scheme is that it has very simple scheme and is easy to be implemented. The main purpose of this paper is to propose and analyze an improved Leap-frog scheme, the so-called continuous-stage modified Leap-frog scheme for high-dimensional semi-linear Hamiltonian wave equations. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Hamiltonian equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula (the Duhamel Principle). Then the continuous-stage modified Leap-frog scheme is formulated. Accordingly, the convergence, energy preservation, symplecticity conservation and long-time behaviour of explicit schemes are rigorously analysed. Numerical results demonstrate the remarkable advantage and efficiency of the improved Leap-frog scheme compared with the existing mostly used numerical schemes in the literature.