  We construct an efficient class of increasingly high-order (up to 17th-order)
essentially non-oscillatory schemes with multi-resolution (ENO-MR) for solving
hyperbolic conservation laws. The candidate stencils for constructing ENO-MR
schemes range from first-order one-point stencil increasingly up to the
designed very high-order stencil. The proposed ENO-MR schemes adopt a very
simple and efficient strategy that only requires the computation of the
highest-order derivatives of a part of candidate stencils. Besides simplicity
and high efficiency, ENO-MR schemes are completely parameter-free and
essentially scale-invariant. Theoretical analysis and numerical computations
show that ENO-MR schemes achieve designed high-order convergence in smooth
regions which may contain high-order critical points (local extrema) and retain
ENO property for strong shocks. In addition, ENO-MR schemes could capture
complex flow structures very well.
