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 4th-order neutrally stable CESE solver of the advection equation du/dt + adu/dx = 0. The space-time stencil of this two-level explicit scheme is formed by one point at the upper time level and three points at the lower time level. Because it is associated with three independent mesh variables it?, (u^)?, and {uxx)V-^s(the numerical analogues of u, du/dx, and d2u/dx2, respectively) and three equations per mesh point, the new scheme is referred to as the a(3) scheme. As in the case of other similar CESE neutrally stable solvers, the a(3) scheme enforces conservation laws in space-time locally and globally, and it has the basic, forward marching, and backward marching forms. 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 o(3) scheme must be neutrally stable when it is stable. Moreover it is proved rigorously that all three amplification factors of the a(3) scheme are of unit magnitude for all phase angles if < 1/2 {y = aAt/Ax). This theoretical result is consistent with the numerical stability condition < 1/2. Through numerical experiments, it is established that the a(3) scheme generally is (i) 4th-order accurate for the mesh variables u? and (ux)?', and 2nd-order accurate for (iixx)?. However, in some exceptional cases, the scheme can achieve perfect accuracy aside from round-off errors.
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