In 2011 a relatively new type of numerical scheme has been introduced: Active Flux schemes. In this type of scheme an extra degree of freedom is added to the cell interfaces of a regular finite volume grid. This enables the use of non-conservative update methods for these additional variables, as conservation is automatically adhered to by the cell integral values. This increases the order of accuracy of the scheme, while allowing a broader range of update methods. This thesis proposes a new update method based on minimizing the truncation error of a Taylor series expansion. This way, a linear update scheme can be created for each unique stencil. An adaptive mesh refinement algorithm is implemented to conform the mesh to local high-frequency phenomena such as shock waves. A high-resolution simulation shows that the adaptive method reaches error levels of a uniform mesh while using ~9.6 times less computational cells.