It depends what you are trying to do exactly. What sort of PDE are you trying to solve? Is it a time-dependent PDE or a just a spatial discretization? LDG works with time-dependent systems, and it seems (from what I'm reading) that it targets non-linear systems. DG is a spatial discretization technique, so you would have to couple it with a time integration scheme to solve time-dependent problems. In my research, I see more focus on the later approach (DG + time integration) as it has great computing performance (FLOPs per second) specially when using high-order discretizations. You can look at MFEM (https://mfem.org/examples/#:~:text=Example%209%3A%20DG%20Advection) for open source code to do just that in a scalable way, or use FEniCS in Python for easier development (albeit, less scalable) https://fenicsproject.org/pub/course/lectures/2015-08-logg-beijing/lecture_10_discontinuous_galerkin.pdf.

Roughly speaking, there are 2 basic approaches for the discontinuous Galerkin (DG) method: modal DG method and nodal DG method. The solution using the modal DG is usually a linear combination of modal coefficients multiplied by a set of orthogonal polynomials u(x)=sum_{k=0:p} c_k*P_k(x) for x in (x_{i-1},x_i), where P_k(x) is the Legendre polynomial of degree k on the interval (x_{i-1},x_i). On the other hand, the nodal DG method utilizes a nodal Lagrange interpolation basis as the approximating basis functions. When using the nodal DG method, the solution is u(x)=sum_{k=0:p} u(r_k)*L_k(x), where L_k is a Lagrange polynomial of degree p using the nodes r_0,..._r_p in [x_{i-1},x_i]. For a general problem, there is preferred method over the other. For example on one advantage of the modal DG method is that the orthogonal polynomials provide a diagonal mass matrix and one advantage of the nodal DG method is that you can choose the nodes r_0,...,r_p as the quadrature points so that you can easily approximate integrals.

If you are working with modal bases, it is more straightforward to change the order of an element. This means that the existing basis functions do not change. Some people enjoy this because it is easier to work with. Some antialiasing techniques require modal bases to be filtered directly, but this is not a performance bottleneck. Modal commands can also highlight sparsity within an element for special operators. This means that the savings are small for the modest order typically used in 3D. Nodal bases simplify the definition of element continuity and simplify the implementation of boundary conditions, contact, and the like. They are also easier to plot, leading to better ḥh-ellipticity in discretized operators. This, in turn, allows the use of less expensive smoothers/preconditioners. This means that people who create ways of solving problems can more easily use terms like "rigid body modes" and "multigrid methods," and they do not have to change how they think about things when they are using these methods. Discretization is a way to turn an ongoing problem into a discrete one that a computer can solve. An embedded discretization is one where the discretization is done in a way that makes it easy to access and use later on.

The main difference between NDG FEM and MDG FEM is in the way they represent the solution to the PDE. In NDG FEM, the solution is represented as a polynomial at each node or element of the mesh, while in MDG FEM, the solution is represented as a modal expansion in terms of a set of basis functions.

Some key points to consider when comparing NDG FEM and MDG FEM: