In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV) method for nonlinear scalar hyperbolic equations in one space dimension. It is well known that the exact solutions to such problems may contain shocks though the initial conditions are smooth, and direct numerical methods may suffer from restricted time step sizes. To relieve the restriction, we propose an ELFV method, where the space-time domain was separated by the partition lines originated from the cell interfaces whose slopes are obtained following the Rakine-Hugoniot junmp condition. Unfortunately, to avoid the intersection of the partition lines, the time step sizes are still limited. To fix this gap, we detect effective troubled cells (ETCs) and carefully design the influence region of each ETC, within which the partitioned space-time regions are merged together to form a new one. Then with the new partition of the space-time domain, we theoretically prove that the proposed first-order scheme with Euler forward time discretization is total-variation-diminishing and maximum-principle-preserving with {at least twice} larger time step constraints than the classical first order Eulerian method for Burgers' equation. Numerical experiments verify the optimality of the designed time step sizes.

翻译：本文针对一维空间非线性标量双曲型方程，构建了一种新颖的欧拉-拉格朗日有限体积法。众所周知，此类问题的精确解即使初始条件光滑也可能包含激波，而直接数值方法常受限于时间步长。为缓解这一限制，我们提出了欧拉-拉格朗日有限体积法：该方法以单元界面为起点生成分割线（其斜率依据Rakine-Hugoniot跳跃条件确定），将时空域进行划分。然而为规避分割线相交，时间步长仍受约束。针对此缺陷，我们通过检测有效问题单元并精心设计每个有效问题单元的影响区域，将区域内分割的时空子域合并为新的统一区域。基于时空域的新划分，我们理论证明：采用欧拉前向时间离散的一阶格式具有总变差衰减性和最大原理保持性，且对于Burgers方程，其时间步长约束比经典一阶欧拉方法宽松至少两倍。数值实验验证了所设计时间步长的最优性。