Implicit time-stepping for advection is applied locally in space and time where Courant numbers are large, but standard explicit time-stepping is used for the remaining solution which is typically the majority. This adaptively implicit advection scheme facilitates efficient and robust integrations with long time-steps while having negligible impact on the overall accuracy, and achieving monotonicity and local conservation on general meshes. A novel and important aspect for the efficiency of the approach is that only one linear solver iteration is needed for each advection solve. The implementation in this paper uses a second-order Runge-Kutta implicit/explicit time-stepping in combination with a second/third-order finite volume spatial discretisation. We demonstrate the adaptively implicit advection in the context of deformational flow advection on the sphere and a fully compressible model for atmospheric flows. Tracers are advected over the poles of latitude-longitude grids with very large Courant numbers and through hexagonal and cubed-sphere meshes with the same algorithm. Buoyant flow simulations with strong local updrafts also benefit from adaptively implicit advection. Stably stratified flow simulations require a stable combination of implicit treatment of gravity and acoustic waves as well as advection in order to achieve long stable time-steps.

翻译：隐式时间步进法仅在库朗数较大的时空区域应用于平流过程，而对通常占多数的其余解域采用标准显式时间步进。这种自适应隐式平流方案在保证整体精度不受显著影响的前提下，实现了在一般网格上的单调性与局部守恒性，并通过长时步推进获得高效稳健的积分结果。该方法效率的一个新颖且重要的特点是：每次平流求解仅需一次线性求解器迭代。本文采用二阶龙格-库塔隐式/显式时间步进与二阶/三阶有限体积空间离散相结合的实现方式。我们通过球面形变流平流和全可压缩大气流动模型展示了自适应隐式平流方案的应用。示踪物在经纬度网格极点处（库朗数极大）的平流传输，以及在六边形网格和立方球面网格上的平流传输，均采用同一算法实现。具有强局部上升气流的浮力流动模拟同样受益于自适应隐式平流方案。为实现长时步稳定计算，稳定分层流动模拟需要将重力波、声波与平流过程进行协同隐式处理。