An Optimal-order Error Estimate to the Modified Method of Characteristics for a Degenerate Convection-diffusion Equation

Time-dependent advection-diffusion equations arise in mathematical models of porous medium flow and transport processes, including petroleum reservoir simulation, environmental modeling, and other applications. In such applications as immiscible displacement of oil by water in a secondary oil recovery process in petroleum industry or a groundwater transport process involving a non-aqueous phase liquid (NAPL), the corresponding governing equation is a degenerate timedependent nonlinear advection-diffusion equation for the saturation of the invading phase. The diffusion in the saturation equation is due to capillary pressure effect, which could vanish or exhibit significant effect depending on whether the wetting phase or the nonwetting phase occupies the pore space [3, 5, 16, 19]. On the other hand, subsurface geological formations often consist of layered media, in which the diffusion parameters could vary by several orders of magnitude. In all of these applications, the governing equations could be convection-dominated in part of the domain while diffusion-dominated in the other part. Consequently, these problems admit solutions with moving fronts and complex structures and present serious mathematical and numerical difficulties. Classical finite difference or finite element methods tend to generate numerical solutions with nonphysical oscillations, while classical upwind methods often produce numerical solutions with excessive numerical diffusion that smears out the fronts and generates spurious effects related to grid orientation [9, 14, 16]. EulerianLagrangian methods provide an alternative approach for numerically solving timedependent advection-diffusion equations. These methods combine the advection and capacity terms in the governing equations to carry out the temporal discretization in the Lagrangian coordinates, and discretize the diffusion term on a fixed mesh in the Eulerian coordinates [6, 11, 13, 24]. They symmetrize the governing equation and stabilize their numerical approximations. Moreover, they generate accurate numerical solutions and significantly reduce the numerical diffusion and