JOURNAL OF COMPUTATIONAL PHYSICS 130, 109–122 (1997) ARTICLE NO. CP965564 Finite Difference Schemes for Three-dimensional Time-dependent Convection-Diffusion Equation Using Full Global Discretization H. Y. Xu, M. D. Matovic, and A. Pollard 1 Centre for Advanced Gas Combustion Technology, Department of Mechanical Engineering, Queen’s University at Kingston, Ontario, Canada K7L-3N6 Received February 20, 1996; revised August 12, 1996 of the discretization is then minimized. This idea is called global discretisation where attention is no longer paid to The three-dimensional, time-dependent convection-diffusion equation (CDE) is considered. An exponential transformation is used the individual spatial and temporal derivatives but to the to collectively transform the CDE. The idea of global discretization whole equation. With this idea, four finite difference is used, where attention is paid to the whole transformed CDE, but schemes are established for both the CDE and the trans- not to the individual spatial and temporal derivatives in the equation. formed CDE. The modified PDEs, following Warming and Four finite difference schemes for both CDE and transformed CDE Hyett [10], are obtained, which indicate that the schemes are established. The modified partial differential equations of these schemes are obtained, which indicate that the trunction errors of are of either second or fourth order if the time step is the schemes can be of second and fourth order, depending on the properly prescribed. A series of analytical solutions to both prescription of the time step length. Some characteristic physical the Navier–Stokes and Burgers equations are chosen as parameters, i.e., local Reynolds number, local Strouhal number, benchmark cases against which the new method is assessed. and viscous diffusive length, are introduced into the schemes and The paper is organized as follows. Sections 2 and 3 demon- the viscous diffusive length is found to be a significant parameter in relating temporal discretisation with spatial discretisation. A se- strate the major steps in establishing the finite difference ries of benchmark analytical solutions of Navier–Stokes and Burgers schemes. Section 4 provides the formulation of the schemes equations, as well as the numerical solutions using the well-known for the CDE and the schemes for the transformed CDE discretisation schemes, are used to investigate the properties of the are given in the Appendix. Section 5 presents the analytical derived schemes. The high-order schemes achieve higher resolu- solutions and the numerical algorithms chosen to bench- tions over the conventional schemes without decreasing much the sparsity of the matrix structures. Grid refinement studies reveal mark the schemes. The numerical experiments are de- that the inverse exponential transformation of the finite difference scribed in Section 6. The results are discussed in Section schemes tends to destroy some resolution of the schemes, espe- 7 and conclusions drawn from the work close out the cially for large local Reynolds number.  1997 Academic Press paper. 2. CONVECTION-DIFFUSION EQUATION 1. INTRODUCTION AND ITS TRANSFORMATION One of the major issues is computational fluid dynamics (CFD) is the discretization of the Navier–Stokes (N-S) Consider the general equation, i.e., the CDE: equations, which are sets of three-dimensional, time- dependent, convection-diffusion equations (CDE). This paper presents a new way to establish finite difference  t + a x  x + a y  y + a z  z = b   2  x 2 +  2  y 2 +  2  z 2  (1) schemes for the CDE, that is, global discretization. The discretization of a partial differential equation (PDE) with + s (t, x, y, z). conventional finite differencing [1] is well known. Typi- cally, discretisation pays attention to individual derivative The coeffecients a x , a y , a z , and b can be reasonably as- terms in the PDE, where the objective is to approximate sumed locally constant. In most existing solution algo- the PDE by replacing it with a set of discretized equations rithms, e.g., SIMPLE [3], PISO [11], and fractional step that are created using some prescribed pattern. Here, the method [12], the pressure derivatives are treated as a PDE is treated in totality and the integral truncation error source term in the momentum equations. Here, it is as- sumed that the source term in Eq. (1) is a function of both 1 To whom correspondence should be addressed. space and time. 109 0021-9991/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.