The CESE development is driven by a belief that a solver should (i) enforce conservation laws in both space and time, and (ii) be built from a non-dissipative (i.e., neutrally stable) core scheme so that the numerical dissipation can be controlled effectively. To provide a solid foundation for a systematic CESE development of high order schemes, in this paper we describe a new high order (4-5th order) and neutrally stable CESE solver of the advection equation ∂u/∂t + a∂u/ ∂x = 0. The space-time

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... stencil of this two-level explicit scheme is formed by one point at the upper time level and two points at the lower time level. Because it is associated with four independent mesh variables u^ , (u . , )^ , (u .,. , ) n j ,and (u .,.,. , ) n j (the numerical analogues of u, ∂u/∂x, ∂ 2 u/∂x 2 , and ∂ 3 u/∂x 3 , respectively) and four equations per mesh point, the new scheme is referred to as the a (4) scheme. As in the case of other similar CESE neutrally stable solvers, the a (4) scheme enforces conservation laws in space-time locally and globally, and it has def the basic, forward marching, and backward marching forms. Assuming |ν| 3 these forms are equivalent and satisfy a space-time inversion (STI) invariant property which is shared by the advection equation. Based on the concept of STI invariance, a set of algebraic relations is developed and used to prove that the a (4) scheme must be neutrally stable when it is stable. Numerically, it has been established that the scheme is stable if |ν| < 1 /3.