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.

翻译：我们构建了一类高效的递增高阶（高达17阶）本质无振荡多分辨率ENO-MR格式，用于求解双曲守恒律。构造ENO-MR格式的候选模板范围从一阶单点模板递增至设计中的极高阶模板。所提出的ENO-MR格式采用了一种极其简单且高效的策略，仅需计算部分候选模板的最高阶导数。除简洁性与高效率外，ENO-MR格式完全无参数且本质上具有尺度不变性。理论分析与数值计算表明，ENO-MR格式在平滑区域（可能包含高阶临界点（局部极值））能达到设计的高阶收敛性，并对强激波保持ENO特性。此外，ENO-MR格式能够极好地捕捉复杂流动结构。