Modified error estimates for discrete variational derivative methods
Many PDEs (for example the Swift–Hohenberg equation and the the KdV equation) have a variational structure. As a direct consequence of this structure we can deduce the existence of properties such as dissipation and conservation laws, as well as the existence of (for example) soliton solutions. A discrete variational derivative method exploits this variational structure when discretising the PDE. In this talk I will use a modified equation analysis to show that the discrete solution of such a method can be considered to be samples of a function which also satisfies a modified variational principle. It is then possible to derive a series of novel conservation or dissipation laws for this new function which mimic those of the original. I will show how these new laws can be constructed and the various consequences of them. This will include the existence of discrete soliton solutions of the KdV equation.