Murli M. Gupta

High accuracy solutions of incompressible Navier-Stokes equations

Abstract In recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type, with particular emphasis on the convection-diffusion equation. These approximations are of compact type, have a local truncation error of fourth order, and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations. In this paper, we extend these high accuracy approximations to the solution of Navier-Stokes equations. Solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors. It is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows.

High accuracy multigrid solution of the 3D convection-diffusion equation

We present an explicit fourth-order compact finite difference scheme for approximating the three-dimensional (3D) convection-diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1. We design a parallelization-oriented multigrid method for fast solution of the resulting linear system using a four-color Gauss-Seidel relaxation technique for robustness and efficiency, and a scaled residual injection operator to reduce the cost of multigrid inter-grid transfer operator. Numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method. Several test problems are solved and highly accurate solutions of the 3D convection-diffusion equations are obtained for small to medium values of the grid Reynolds number. Effects of using different residual projection operators are compared on both vector and serial computers.

Nature of viscous flows near sharp corners

Abstract Explicit solutions of two-dimensional, steady-state Navier-Stokes equations are derived in the neighborhood of sharp corners where a sliding wall meets a stationary wall and causes a mathematical singularity. These solutions are valid for small Reynolds numbers. A semi-analytic technique is used to derive these solutions. Some comparisons with numerical solutions are also carried out.

Some Difference Schemes for the Biharmonic Equation

The Dirichlet problem for biharmonic equation in a rectangular region is considered. The method of splitting is used and two classes of finite difference approximations are defined. Two semi-iterative procedures are considered for obtaining the solution of the resulting coupled system of algebraic equations. It is shown that the rate of convergence of the iterative procedures depends upon the choice of the difference approximation. Estimates for optimum iteration parameters are given and several comparisons are made. An attempt is made to unify the ideas on the splitting technique for solving the first biharmonic boundary value problem.

Convergence of Fourth Order Compact Difference Schemes for Three-Dimensional Convection-Diffusion Equations

We consider a Dirichlet boundary-value problem for the three-dimensional convection-diffusion equations with constant coefficients in the unit cube. A high order compact finite difference scheme is constructed on a 19-point stencil using the Steklov averaging operators. We prove that the finite difference scheme converges in discrete

Discretization Error Estimates for Certain Splitting Procedures for Solving First Biharmonic Boundary Value Problems

W_2^m(\omega)

Direct solution of the biharmonic equation using noncoupled approach

-norm with the convergence rate

High accuracy solution of three-dimensional biharmonic equations

O(h^{s-m})

A comparison of numerical solutions of convective and divergence forms of the Navier-Stokes equations for the driven cavity problem

, where the real parameter

On the improvement of convergence rate of difference schemes with high order differences for a convection-diffusion equation

s