Numerical solution of the three-dimensional advection–diffusion equation

• Published in: Applied mathematics and computation Vol. 150; no. 1; pp. 5 - 19
• Main Author: Dehghan, Mehdi
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 27.02.2004; Elsevier
• Subjects: Advection–diffusion processes; Advective terms; Difference and functional equations, recurrence relations; Diffusive derivatives; Exact sciences and technology; Finite difference techniques; Fully explicit methods; Fully implicit schemes; Mathematics; Modified equivalent partial differential equations; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Stability; Fully implicit schemes; Stability; Fully explicit methods; Modified equivalent partial differential equations; Advective terms; Diffusive derivatives; Finite difference techniques; Advection–diffusion processes; Numerical analysis; Implicit method; Fluid dynamics; Explicit method; Numerical solution; Advection diffusion equation; Diffusive derivative; Partial differential equation; Numerical stability; Finite difference method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/S0096-3003(03)00193-0

The study of advection–diffusion equation continues to be an active field of research. The subject has important applications to fluid dynamics as well as many other branches of science and engineering. In this paper several different numerical techniques will be developed and compared for solving the three-dimensional advection–diffusion equation with constant coefficient. These techniques are based on the two-level fully explicit and fully implicit finite difference approximations. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. Another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties. The new second-order accurate methods are free of numerical diffusion. The results of a numerical experiment are presented, and the accuracy and central processor (CPU) time needed are discussed and compared.