Finite Difference Schemes for Three-dimensional Time-dependent Convection-Diffusion Equation Using Full Global Discretization

Abstract The three-dimensional, time-dependent convection-diffusion equation (CDE) is considered. An exponential transformation is used to collectively transform the CDE. The idea of global discretization is used, where attention is paid to the whole transformed CDE, but not to the individual spatial and temporal derivatives in the equation. Four finite difference schemes for both CDE and transformed CDE are established. The modified partial differential equations of these schemes are obtained, which indicate that the trunction errors of the schemes can be of second and fourth order, depending on the prescription of the time step length. Some characteristic physical parameters, i.e., local Reynolds number, local Strouhal number, and viscous diffusive length, are introduced into the schemes and the viscous diffusive length is found to be a significant parameter in relating temporal discretisation with spatial discretisation. A series of benchmark analytical solutions of Navier–Stokes and Burgers equations, as well as the numerical solutions using the well-known discretisation schemes, are used to investigate the properties of the derived schemes. The high-order schemes achieve higher resolutions over the conventional schemes without decreasing much the sparsity of the matrix structures. Grid refinement studies reveal that the inverse exponential transformation of the finite difference schemes tends to destroy some resolution of the schemes, especially for large local Reynolds number.