Remarks on energy methods for structure-preserving finite difference schemes - Small data global existence and unconditional error estimate

In the previous article (Yoshikawa, 2017), the author proposes the energy method for structure-preserving finite difference schemes, which enable us to show global existence and uniqueness of solution for the schemes and error estimates. In this article, we give two extended remarks of the methods. One is related to the small data global existence results for schemes of which energy is not necessarily bounded from below. The other is an unconditional error estimate which holds globally in time and without smallness condition for split sizes. These results can be shown due to the structure-preserving property.