Fourth‐order exponential finite difference methods for boundary value problems of convective diffusion type

Abstract

SUMMARYMethods based on exponential finite difference approximations of h 4 accuracy are developed to solve one and two-dimensional convection-diffusion type differential equations with constant and variable convection coefficients. In the one-dimensional case, the numerical scheme developed uses three points. For the two-dimensional case, even though nine points are used, the successive line overrelaxation approach with alternating direction implicit procedure enables us to deal with tri-diagonal systems. The methods are applied on a number of linear and non-linear problems, mostly with large first derivative terms, in particular, fluid flow problems with boundary layers. Better accuracy is obtained in all the problems, compared with the available results in the literature. Application of an exponential scheme with a non-uniform mesh is also illustrated. The h 4 accuracy of the schemes is also computationally demonstrated.

High-order compact exponential finite difference methods for convection–diffusion type problems

“…In addition, scheme (6) provides the exact solution for the 1D convection-diffusion equation with constant convection coefficient in the absence of a source term. Scheme (6) and its some variants have been proposed via other approaches [1,[27][28][29].…”

“…Thus, the existing HOC polynomial FD schemes are not suitable for particular physical problems, such as abrupt boundary layer in convection-dominated problems and shock-like discontinuities caused by local nonlinearities, unless a very fine mesh is used [19,20]. This dilemma can be resolved by utilizing nonuniform mesh and local mesh refinement strategies [4,5,12,13,27,30]. Unfortunately, the boundary layer location or the singularity region must be known in this case.…”

“…However, it is easily found that the influence of source term in the FD equation established in [3] becomes less and less as the cell Reynolds number Re c increased, and eventually for very large Re c value, this influence becomes negligible. The drawbacks of the exponential FD methods proposed in [3,27] will be illustrated in detail in the numerical examples.…”

“…MacKinnon and Johnson [24] derived an 4OC exponential FD scheme for the 2D convection-diffusion equation with constant convection coefficients. Recently, Radhakrishna Pillai [27] developed 4OC exponential FD methods for solving 1D and 2D convection-diffusion equations with constant and variable convection coefficients on compact stencils. However, the coefficient matrix of schemes may fail to be diagonally dominant for the convection-dominated problems with variable convection coefficients, since the influence coefficients involve both exponential and polynomial functions in the FD approximations.…”

“…The UD scheme is unconditional stable, but is only first-order accurate, and the resulting solution exhibits the effect of artificial viscosity [28,30]. In recent years, high order compact (HOC) FD methods have generated renewed interest and a variety of specialized techniques have been developed [3][4][5][6]9,15,17,19,20,[22][23][24]27,[32][33][34]37]. For the 1D convectiondiffusion equations, Dekema and Schultz [6] developed higher-order methods using the Taylor series expansion.…”