Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations

The paper considers the numerical solution of boundary-value problems for multidimensional convection-diffusion type equations (CDEs). Such equations are useful for various physical processes in solids, liquids and gases. A new approach to the spatial approximation for such equations is proposed. This approach is based on an integral transformation of second-order one-dimensional differential operators. A linear version of CDE was chosen for simplicity of the analysis. In this setting, exponential difference schemes were constructed, algorithms for their implementation were developed, a brief analysis of the stability and convergence was made. This approach was numerically tested for a two-dimensional problem of motion of metallic particles in water flow subject to a constant magnetic field.     

Tridiagonal solver-based L-stable one-step schemes for nonlinear parabolic equations

An attractive feature of the widely used Crank-Nicolson (C-N) scheme for parabolic equations is that it is a tridiagonal solver-based (TSB) scheme. But, in case of inconsistencies in the initial and boundary conditions or when the ratio of temporal to spatial steplengths is large, it can produce unwanted oscillations or an unacceptable solution. As alternative to C-N, Chawla et al. [2, 3] introduced L-stable generalized trapezoidal formulas (GTF(α)) which can give a more acceptable solution by a judicious choice of the parameter α; however, GTF are not TSB schemes. It is natural to ask for L-stable TSB schemes. In the present paper, we first introduce a one-parameter family of generalized midpoint formulas (GMF(α)); again GMF are not TSB schemes. We then introduce a two-parameter family through a linear combination of the GMF and the classical trapezoidal formula, and show the existence of a one-parameter subfamily of L-stable TSB schemes; these schemes are unconditionally stable. The computational performance of the obtained schemes is compared with the C-N scheme by considering a nonlinear reaction-diffusion equation.     

Exponential difference schemes with double integral transformation for solving convection-diffusion equations

Convection-diffusion equations are studied. These equations are used for describing many nonlinear processes in solids, liquids, and gases. Although many works deal with solving them, they are still challenging in terms of theoretical and numerical analysis. In this work, the grid approach based on the method of finite differences for solving equations of this kind is considered. In order to make it easier, the one-dimensional version of such an equation was chosen. However, the equation preserves its principal properties; i.e., it is non-monotonic and non-linear. To solve boundary-value problems for such equations, a special variant of the non-monotonic sweep procedure is proposed.     

Extended one-step methods for the numerical solution of ordinary differential equations

A class of one-step finite difference formulae for the numerical solution of first-order differential equations is considered. The accuracy and stability properties of these methods are investigated. By judicious choice of the coefficients in these formulae a method is derived which is both A-stable and third-order convergent. Moreover the new method is shown to be L-stable and so is appropriate for the solution of certain stiff equations. Numerical results are presented for several test problems.     

On the extended one-step schemes for solving stiff systems of ordinary differential equations

Another fourth order extended one-step implicit scheme of solving stiff ordinary differential equations is introduced in this paper, through which, it is shown that such schemes are literally classical implicit Runge-Kutta schemes. Using general theory of Runge-Kutta schemes, stabilities other than A-stability or L-stability are further investigated for the proposed scheme. It is also shown that the parameters involved in such schemes can be better used to reduce the computation cost, making such schemes thus more competitive with traditional ones. Numerical examples are presented showing the competence of such schemes in solving a variety of stiff systems.     

Extended visual cryptography schemes

Visual cryptography schemes have been introduced in 1994 by Naor and Shamir. Their idea was to encode a secret image into n shadow images and to give exactly one such shadow image to each member of a group P of n persons. Whereas most work in recent years has been done concerning the problem of qualified and forbidden subsets of P or the question of contrast optimizing, in this paper we study extended visual cryptography schemes, i.e., shared secret systems where any subset of P shares its own secret.     

Moment preserving schemes for Euler equations

A high order accurate finite difference scheme is proposed for one-dimensional Euler equations. In the scheme a set of first three moments of each signal are preserved during the updating. The scheme is one of 5th order in space and 4th order in time. This feature is different from that in typical existing methods in which the use of the first three polynomials results in only 3rd order accuracy in space. The scheme has different features from the existing high order schemes, and the most noticeable are the simultaneous discretization both in space and time, and the use of moments of Riemann invariants instead of primitive physical variables. Numerical examples are given to show the accuracy of the scheme and its robustness for the flows involving shocks.     

A one-step subregion method for delay differential equations

We study a one-step method for delay differential equations, which is equivalent to an implicit Runge-Kutta method. It approximates the solution in the whole interval with a piecewise polynomial of fixed degree n. For an appropiate choice of the mesh points, it provides uniform convergence 0(hⁿ⁺¹) and the superconvergence 0(h²ⁿ) at the nodes.     

Coefficients for studying one-step rational schemes for IVPs in ODEs: I

In [1,2], we considered an approach of global formulation of one-step integrators of arbitrary order on a self-adjusting rational approximant by introducing the constants C^(s)_j, s = 1, 2, 3, …, j = 2, 3, …, j − s ≥ 1. In the present consideration, we motivate a study of this approach to derive a class of one-step methods of variable order and variable stepsize.     

Compact high-order schemes for the Euler equations

An implicit approximate factorization (AF) algorithm is constructed, which has the following characteristics.

Monotonic bicompact schemes for linear transport equations

The biocompact difference scheme earlier proposed by these authors for a linear transport equation, which has the fourth-order approximation in spatial coordinate on the two-point stencil and the first-order approximation in time, is monotonic. This implicit scheme is absolutely stable and can be solved by explicit formulas of a running calculation. On the basis of this scheme a monotone non-linear homogeneous difference scheme of high (third for smooth solutions) order accuracy in time is constructed. Calculations of test problems with discontinuous solutions have demonstrated that the proposed scheme has a significant advantage in accuracy over the known nonoscillatory schemes of high-order approximation.     

Multidimensional exponentially fitted simplicial finite elements for convection-diffusion equations with tensor-valued diffusion

Abstract In this paper we propose and analyze an exponentially fitted simplicial finite element method for the numerical approximation of solutions to diffusion-convection equations with tensor-valued diffusion coefficients. The finite element method is first formulated using exponentially fitted finite element basis functions constructed on simplicial elements in arbitrary dimensions. Stability of the method is then proved by showing that the corresponding bilinear form is coercive. Upper error bounds for the approximate solution and the associated flux are established.     

An explicit finite element method for convection-diffusion equations using rational basis functions

An explicit Galerkin method is formulated by using rational basis functions. The characteristics of the rational difference scheme are investigated with regard to consistency, stability and numerical convergence of the method. Numerical results are also presented.     

Nonlinear difference schemes for linear partial differential equations

We discuss a nonlinear difference scheme for approximating the solution of the initial value problem for linear partial differential equations. At each time step of the calculation the method proceeds by processing the data and determining the best possible scheme to use for that step, according to an optimization criterion to be described. We show that the method is stable and convergent applicating it on the heat equation. In all cases considered the nonlinear method was more accurate than the classical methods.     

Conservative upwind difference schemes for the Euler equations

In a recent paper [1] a number of numerical schemes for the shallow water equations based on a conservative linearization are analyzed. In particular, it is established that the schemes are related through the use of a source term. In this paper this technique is applied to the Euler equations, and further analysis suggests a new formulation of an existing scheme having the same key properties.     

A test problem for numerical schemes for nonlinear hyperbolic equations

In this note we solve a model problem to test the applicability of numerical schemes for hyperbolic partial differential equations with shock-containing solutions. This test problem was used successfully in [1]in order to compare analytic and numerical solutions. The construction is made possible by considering the development of a jump discontinuity in the solution of scalar quasilinear equations with piecewise smooth data. This is a simple generalization of a discussion by Lax [2].