  In this paper, we develop a new discontinuous Galerkin method for solving
several types of partial differential equations (PDEs) with high order spatial
derivatives. We combine the advantages of local discontinuous Galerkin (LDG)
method and ultra-weak discontinuous Galerkin (UWDG) method. Firstly, we rewrite
the PDEs with high order spatial derivatives into a lower order system, then
apply the UWDG method to the system. We first consider the fourth order and
fifth order nonlinear PDEs in one space dimension, and then extend our method
to general high order problems and two space dimensions. The main advantage of
our method over the LDG method is that we have introduced fewer auxiliary
variables, thereby reducing memory and computational costs. The main advantage
of our method over the UWDG method is that no internal penalty terms are
necessary in order to ensure stability for both even and odd order PDEs. We
prove stability of our method in the general nonlinear case and provide optimal
error estimates for linear PDEs for the solution itself as well as for the
auxiliary variables approximating its derivatives. A key ingredient in the
proof of the error estimates is the construction of the relationship between
the derivative and the element interface jump of the numerical solution and the
auxiliary variable solution of the solution derivative. With this relationship,
we can then use the discrete Sobolev and Poincar\'{e} inequalities to obtain
the optimal error estimates. The theoretical findings are confirmed by
numerical experiments.
