Description
In this talk, we consider an energy-preserving finite difference
scheme for the KdV equation (Furihata, 1996). The scheme preserves
the cubic energy function, and has been empirically known that
it works better than (L2)norm-preserving schemes. However,
since the cubic energy function by itself is not useful
in the mathematical analysis of numerical schemes,
the convergence estimate of the scheme has been left open until now
(in contrast to the norm-preserving cases, where various results
are known.)
Recently, we devised a new convergence estimate argument that
successfully gives the desired estimate. The argument is constructed
in an inductive way, and can be applied to the cases where
useful a priori estimates on the solution (such as in L2 or sup)
are hard to obtain. It can be applied to other energy-preserving
schemes for the generalized KdV or the Ostrovsky equation.
