A priori error estimates for upwind finite volume schemes for two-dimensional linear convection diffusion problems

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in R² and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h^1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.     

Inverse problem for a class of one-dimensional wave equations with piecewise constant coefficients

We consider the problem of reconstructing the piecewise constant coefficient of a one-dimensional wave equation on the halfline from the knowledge of the displacement on the boundary caused by an impulse at time zero. This problem is formulated as a nonlinear optimization problem. The objective function of this optimization problem has several special features that have been exploited in building an ad hoc optimization method. The optimization method is based on the solution of a nonlinear system of equations by an algorithm consisting of the evaluation of the unknowns one by one.The research of the third author has been made possible through the support and sponsorship of the Italian Government through the Ministero Pubblica Istruzione under Contract M.P.I. 60% 1987 at the Università di Roma—La Sapienza.     

Exact finite difference scheme for linear differential equation with constant coefficients

An exact finite difference equation for the n-th order linear differential equation with real, constant coefficients is constructed. The exact finite difference scheme is expressed differently but equivalent to that given by Potts [3].     

Time-global smoothing estimates for a class of dispersive equations with constant coefficients

We discuss smoothing effects of homogeneous dispersive equations with constant coefficients. In the case where the characteristic root is positively homogeneous, time-global smoothing estimates are known. It is also known that a dispersiveness condition is necessary for smoothing effects. We show time-global smoothing estimates where the characteristic root is not necessarily homogeneous. Our results give a sufficient condition so that lower order terms can be absorbed by the principal part, and also indicate that smoothing effects may be caused by lower order terms in the case where the dispersiveness condition fails to hold.     

The FD method for first-order linear hyperbolic differential equations with piecewise smooth coefficients

A justification of the differential-discrete FD method for solving one class of first-order linear partial differential equations with piecewise smooth coefficients, a right-hand side, and initial conditions is presented. The convergence rate of the method is shown to be limited exclusively by the smoothness of the input information. High accuracy, ease of algorithmic implementation, and feasibility of representation of results in an analytical form point to the advantage of the FD method over the other methods for the given class of problems. Bibliography:6 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp. 1–11     

A new method to solve the second-order linear difference equations with constant coefficients

In this paper, we give a new and straightforward method to solve the non-homogeneous second-order linear difference equations with constant coefficients. It is new because it does not require the uniqueness theorem of the solution of the problem of initial values. Neither does it require a fundamental system of solutions, nor the method of variation of parameters. Moreover, we get a unique formula that expresses the general solution independently of the multiplicities of the roots of the characteristic equation.     

Campanato type estimates for solutions of difference-elliptic partial differential equations with constant coefficients

Difference-elliptic partial differential equations are discussed and Campanato type estimates are obtained for solutions of the equations.     

Campanato type estimates for solutions of difference-elliptic partial differential equations with constant coefficients

Difference-elliptic partial differential equations are discussed and Campanato type estimates are obtained for solutions of the equations.     

Method for constructing solutions of linear ordinary differential equations with constant coefficients

For Cauchy problems involving linear differential equations with constant coefficients, a new method for constructing solutions without determining the roots of the characteristic equation is proposed. Formulas for the differentiation of the solution with respect to the equation coefficients are derived, and an approximate analytical solution is found.     

On the derivation of new mean-value formulas for linear differential equations with constant coefficients

In the framework of a symbolic approach to mean-value formulas, we suggest a method for the derivation of new mean-value formulas for some classes of partial differential equations.     

A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients

In recent years, there has been considerable interest in extending the well-known Calderon-Zygmund estimates for the Laplacian to more general equations, in particular equations with highest order coefficients lying in the Sarason space VMO. In addition, the analogous estimates with Morrey spaces replacing Lebesgue spaces have been considered. These Morrey space estimates have been proved by refining the proofs for the L^p estimates. We shall show that the Morrey space estimates follow from the L^p estimates via an elementary argument which is very similar to that used by Campanato.     

A spline collocation method for singular integral equations with piecewise continuous coefficients

We consider the collocation method with piecewise linear trial functions for systems of singular integral equations with Cauchy kernel and piecewise continuous coefficients. Necessary and sufficient conditions for the stability in L² are given. The results are obtained in the case of a closed Ljapunov curve as well as in the case of an interval. The proof of the main theorem is based on a modification of the Banach algebra technique established in the local principle by Gohberg and Krupnik [2]. Our results extend those obtained by Prößdorf and Schmidt [9, 10] from the case of continuous coefficients and unit circle to the case of piecewise continuous coefficients.     

A simplex-type algorithm for continuous linear programs with constant coefficients

Mathematical Programming - We consider continuous linear programs over a continuous finite time horizon T, with a constant coefficient matrix, linear right hand side functions and linear cost...     

On the multisummability of divergent solutions of linear partial differential equations with constant coefficients

We consider the Cauchy problem for general linear partial differential equations in two complex variables with constant coefficients. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of the Cauchy data.     

Arbitrary-order difference schemes for solving linear advection equations with constant coefficients by the Godunov method with antidiffusion

An approach to the construction of second-and higher order accurate difference schemes in time and space is described for solving the linear one-and multidimensional advection equations with constant coefficients by the Godunov method with antidiffusion. The differential approximations for schemes of up to the fifth order are constructed and written. For multidimensional advection equations with constant coefficients, it is shown that Godunov schemes with splitting over spatial variables are preferable, since they have a smaller truncation error than schemes without splitting. The high resolution and efficiency of the difference schemes are demonstrated using test computations.