A stable numerical method for solving variable coefficient advection–diffusion models

In a recent paper [E. Defez, R. Company, E. Ponsoda, L. Jódar, Aplicación del método CE-SE a la ecuación de advección-difusión con coeficientes variables, Congreso de Métodos Numéricos en Ingenierá (SEMNI), Granada, Spain, 2005] a modified space–time conservation element and solution element scheme for solving the advection–diffusion equation with time-dependent coefficients, is proposed. This equation appears in many physical and technological models like gas flow in industrial tubes, conduction of heat in solids or the evaluation of the heating through radiation of microwaves when the properties of the media change with time. This method improves the well-established methods, like finite differences or finite elements: the integral form of the problem exploits the physical properties of conservation of flow, unlike the differential form. Also, this explicit scheme evaluates the variable and its derivative simultaneously in each knot of the partitioned domain. The modification proposed in [E. Defez, R. Company, E. Ponsoda, L. Jódar, Aplicación del método CE-SE a la ecuación de advección-difusión con coeficientes variables, Congreso de Métodos Numéricos en Ingenierá (SEMNI), Granada, Spain, 2005] with regard the original method [S.C. Chang, The method of space–time conservation element and solution element. A new approach for solving the Navier–Stokes and Euler equations, J. Comput. Phys. 119 (1995) 295–324] consists of keeping the variable character of the coefficients in the solution element, without considering the linear approximation. In this paper the stability of the proposed method is studied and a CFL condition is obtained.