Nonlinearly-Stable High-Order Methods on Simplices with Improved Efficiency

In one dimension, integration by parts is represented as follows. We replicate this process in a discrete setting, which allows us to achieve the desired property. It is important to note that this approach does not lead to truncation error; rather, it holds true for any mesh, similar to the principle of conservation. Thus, conservation is maintained not only in the limit of mesh refinement but also on a finite mesh.

In this context, we observe that while the individual components do not exactly approximate the desired result, their summation yields the correct outcome. Here, D represents the derivative operator, H approximates a volume integral, and E approximates a surface integral. In one dimension, the surface integral is less apparent as it corresponds to just two points. However, in two dimensions, the integration by parts becomes more complex, and the surface integral is clearly represented by E.