SUMMARYA new finite differenoe scheme for the convection-diffusion equation with variable coefficients is proposed. The difference scheme is defined on a single square cell of size 2h over a 9-point stencil and has a truncation error of order h4. The resulting system of equations can be solved by iterative methods. Numerical results of some test problems are given.

In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9-point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated naturally. Since the approximations use the 9-point compact stencil, no special formulas are needed near the boundaries. Both second-order and fourth-order discretizations are derived.The fourth-order approximations produce more accurate results than the 13-point classical stencil or the commonly used system of two second-order equations coupled with the boundary condition.The method suffers from slow convergence when classical iteration methods such as Gauss-Seidel or SOR are employed. In order to alleviate this problem we propose several multigrid techniques that exhibit grid-independent convergence and solve the biharmonic equation in a small amount of computer time. Test results from three different problems, including Stokes flow in a driven cavity, are reported.

A high-order finite-difference approximation is proposed for numerical solution of linear or quasilinear elliptic differential equations. l h e approximation is defined o n a square mesh stencil using nine node points and has a truncation error of order 11'. Several test problems. including one modeling convection-dominated flows. arc solved using this and existing methods. The results clearly exhibit the superiority of the new approximation, in terms of both accuracy and computational efficiency.

This paper describes an introductory course in discrete structures for the undergraduate computer science student that has evolved at the University of Saskatchewan, Saskatoon. The philosophy of suth a course and certain problems encountered in preparing and teaching it are discussed.

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