Higher order compact scheme for convection-diffusion equations

Abstract

Differential equations arise in almost all areas of science and engineering including fluid dynamics, fluid mechanics, aerodynamics, medicines, whether predictions etc. Most of the realistic model problems are defined in terms of differential equations or system of differential equations. But most of these model problems or differential equations are not solvable by analytical methods because of the complexity of the coefficients or nonlinearity or due to complex domains etc. Therefore, we need to depend on numerical techniques to find the approximate solutions of the differential equations. In the present study, higher order compact finite difference schemes has been presented for solving the differential equations. A brief outline of the thesis is as follows:\\

Chapter 1 includes the basic definitions and concepts related to the differential equations. Types of differential equations and solution methodology has been discussed. Analytical and numerical schemes has been explained with the help of examples. \\

In Chapter 2, a fourth order compact scheme has been proposed for a convection-diffusion differential equation with variable coefficients. In the derivation of the difference scheme, the solution u(x,y) is first expressed locally on a mesh element in terms of a linear combination of polynomial functions. The coefficients p(x,y), q(x,y), and f(x,y) are expended in a similar manner. A set of linear equations for the unknown coefficients in the expansion of u(x,y) are obtained by demanding that the governing differential equation be satisfied locally at each mesh point. Additional equations are obtained by interpolating the solution over a set of mesh points which lie on the cell. The difference scheme is defined on a single square cell of size 2h over a 9-point stencil. Only those mesh points which lie on a single square cell of side 2h are involved, thereby keeping the bandwidth as small as possible for the order of the truncation error achieved. The proposed compact scheme has a truncation error of order h^{4}. The resulting scheme has been applied to a flow problem in porous media.\\

In Chapter 3, a fourth order compact finite difference scheme for a second order ordinary differential equation has been presented. To formulate the proposed scheme, the second-order differential equation has been supplemented with fourth order relations between the function value y and the spatial first and second derivatives, F(=y^{'}) and S(=y^{''}), on three adjacent mesh points and the derivatives F and S has been eliminated from the system. In this way, a scheme of fourth order, yet based on only three points, is constructed. A proper combination of fourth order relations with boundary conditions and the differential equation ensures a fourth order truncation error also at the boundaries. The scheme is based on a non-equidistant mesh, which makes it particularly useful in problems involving shard boundary layers.

Bibliography has been presented in the last.