The structural method, introduced over the last two years, develops high-order numerical schemes for solving partial differential equations (PDEs) on compact stencils. This finite difference approach is unique in that it approximates both the solution and its derivatives with the same order of accuracy by defining two independent sets of discrete equations: the physical equations (PEs) and the structural equations (SEs). The PEs represent the problem's physics, while SEs ensure the accuracy of the discretization. This separation provides a high degree of flexibility, allowing for the modification of PEs to include specific constraints (e.g., ensuring a vector field is divergence-free) and the adjustment of SEs to handle non-smooth solutions or enhance stability. The ANR project SMEAGOL seeks to extend this method to hyperbolic systems of balance laws in multiple spatial dimensions, where continuous initial conditions often lead to non-smooth solutions and multiscale regime changes. This makes the method particularly suitable for complex problems in fluid mechanics or electromagnetism. The separation between physical and structural equations in this framework allows for the dynamic adaptation of the scheme to the local problem, switching PEs or SEs on or off as needed. SMEAGOL's goals include both constructing and adapting the structural method to these advanced applications, to develop well-balanced, asymptotic-preserving and robust schemes.