In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme.

翻译：本文针对对流扩散方程，发展了高阶、守恒、非分裂的欧拉-拉格朗日(EL)龙格-库塔(RK)有限体积(FV)加权本质无振荡(WENO)格式。所提出的EL-RK-FV-WENO格式通过定义修正特征线并沿其特征线演化解，显著放松了对流项的时间步长约束。算法设计的主要挑战源于在动态变化的拉格朗日网格上构建精确且稳健的重构过程的复杂性。该重构过程对于时变上游四边形上的通量计算以及沿运动特征线的时间积分是必需的。为解决这一问题，我们提出一种策略：在固定欧拉网格上采用WENO重构进行空间重构，并在欧拉背景网格上更新隐显式RK时间积分的中间解。该策略利用高效的重构与重映射算法处理时变四边形上多项式重构的复杂性，同时保证局部质量守恒。由于通量形式的半离散化以及在背景网格与上游网格上均采用质量守恒重构，所提格式确保了质量守恒。大量数值实验验证了该格式的有效性。