A novel discrete variational derivative method using ``average-difference methods''

<p> We consider conservative numerical methods for a certain class of PDEs, for which standard conservative methods are not effective. There, the standard skew-symmetric difference operators indispensable for the discrete conservation law cause undesirable spatial oscillations. In this letter, to circumvent this difficulty, we propose a novel ``average-difference method,'' which is tougher against such oscillations. However, due to the lack of the apparent skew-symmetry, the proof of the discrete conservation law becomes nontrivial. In order to illustrate partially the superiority, we compare the standard and proposed methods for the linear Klein--Gordon equation. </p>