An Energy Conservative Numerical Scheme on Mixed Meshes for the Nonlinear Schrodinger Equation

As is well known, for PDEs that enjoy conservation properties, numerical schemes that inherit the properties are advantageous in that the schemes give qualitatively better solutions in practice. Lately Furihata and Matsuo have developed "the discrete variational method" that automatically constructs conservative finite difference schemes on uniform meshes for a class of PDEs with certain variational structures. We extend this method to mixed meshes and derive a numerical scheme that conserves the energy and the density for the nonlinear Schrodinger equation on such meshes.