AN IMMERSED EULERIAN-LAGRANGIAN LOCALIZED ADJOINT METHOD FOR TRANSIENT ADVECTION-DIFFUSION EQUATIONS WITH INTERFACES

Transient advection-diffusion equations arise in mathematical models for describing petroleum reservoir simulation, groundwater contaminant transport, geological storage of carbon dioxide and remediation, and many other applications [1, 13, 2, 7, 12, 13, 14, 20]. These equations admit solutions with moving steep fronts and complicated structures. Furthermore, subsurface porous medium matrix often contains a variety of faults and fractures of different magnitude. Those relatively large faults must be accurately incorporated into the corresponding mathematical models, in which the geological formations consist of several subdomains with different geological properties and salient physical interfaces. This also means that in the numerical discretization the computational meshes must align with the large faults in order to obtain a stable and accurate numerical solution. Note that the number of large faults is usually quite limited, so the modeling and numerical implementation is doable. On the other hand, there are numerous relatively small fractures which are very difficult, if not impossible at all, to describe in a deterministic manner geologically. As a matter of fact, these relatively tiny fractures are often described in a probability sense. The impact of these tiny fractures can be handled via the approach of upscaling or multiscale numerical techniques. As for those intermediate fractures, they are probably too big to be upscaled into the underlying numerical schemes in any reasonable manner. On the other hand, there are probably too many intermediate fractures such that the computational meshes of the underlying numerical scheme align with each of them. Based on these considerations we plan to adopt the approach of immersed numerical method to handle these intermediate fractures. To expose the idea, in this paper we consider the one-dimensional transient linear advection-diffusion equation with interfaces

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