Many systems can be described as two or more domains of distinct phases or compositions separated by a thin interface. Modeling these systems is difficult because it often requires solving differential equations with boundary conditions on complicated or irregular boundaries. The smoothed-boundary method is a numerical technique that circumvents these problems by describing interfaces as diffuse regions where a phase-field-like parameter varies smoothly between values defining separate domains. This powerful method allows us to solve PDEs within boundaries of arbitrary geometry; however, a high-resolution is needed to accurately describe very thin interfacial regions, which on a uniform grid becomes computationally expensive. In this project, we develop an algorithm for generating an adaptive Cartesian mesh for the smoothed-boundary method, with the goal of increasing the accuracy and speed of numerical simulations. We validate our algorithm against benchmark simulations applying the smoothed-boundary method on a fixed uniform mesh, and compare the accuracy and computational requirements to simulations performed using or adaptive meshing technique. In addition, we demonstrate how our methods can be applied to problems in fluid dynamics.