The paper introduces the "lymph" library, a MATLAB-based tool for solving multi-physics differential problems using high-order discontinuous Galerkin (DG) methods on polytopal grids. The library is designed to handle complex geometries and heterogeneous physical parameters, offering high-order accuracy, geometric flexibility, and robustness. It supports both spatial and temporal discretization, with spatial discretization based on polytopal meshes and time integration using schemes like Crank-Nicolson and Newmark-β. The library includes functions for mesh generation, finite element space construction, quadrature formulas, and post-processing. It is structured into core and physics folders, with the core handling general DG discretization and physics folders containing specific solvers for problems like the Poisson equation, heat equation, and elastodynamics. The library is flexible, allowing coupling of different physics and implementation of new ones. The paper demonstrates the library's capabilities through examples, showing convergence properties and performance on various problems, including the Poisson equation, heat equation, and elastodynamics. The library is open-source and provides a user guide for implementation and solution of differential problems. It is also compatible with existing mesh generation tools and can be extended with machine learning techniques for mesh refinement and coarsening. The paper highlights the advantages of the PolyDG method, including its ability to handle complex geometries, non-conforming interfaces, and high-order accuracy. The library is tested on various problems, demonstrating its effectiveness in solving multi-physics differential problems with high-order DG methods on polytopal grids.The paper introduces the "lymph" library, a MATLAB-based tool for solving multi-physics differential problems using high-order discontinuous Galerkin (DG) methods on polytopal grids. The library is designed to handle complex geometries and heterogeneous physical parameters, offering high-order accuracy, geometric flexibility, and robustness. It supports both spatial and temporal discretization, with spatial discretization based on polytopal meshes and time integration using schemes like Crank-Nicolson and Newmark-β. The library includes functions for mesh generation, finite element space construction, quadrature formulas, and post-processing. It is structured into core and physics folders, with the core handling general DG discretization and physics folders containing specific solvers for problems like the Poisson equation, heat equation, and elastodynamics. The library is flexible, allowing coupling of different physics and implementation of new ones. The paper demonstrates the library's capabilities through examples, showing convergence properties and performance on various problems, including the Poisson equation, heat equation, and elastodynamics. The library is open-source and provides a user guide for implementation and solution of differential problems. It is also compatible with existing mesh generation tools and can be extended with machine learning techniques for mesh refinement and coarsening. The paper highlights the advantages of the PolyDG method, including its ability to handle complex geometries, non-conforming interfaces, and high-order accuracy. The library is tested on various problems, demonstrating its effectiveness in solving multi-physics differential problems with high-order DG methods on polytopal grids.