This paper is devoted to the study of the Eulerian-Lagrangian method (ELM) for convection-diffusion equations on unstructured grids with or without accurate numerical integration. We first propose an efficient and accurate algorithm to calculate the integrals in the Eulerian-Lagrangian method. Our approach is based on an algorithm for finding the intersection of two non-matching grids. It has optimal algorithmic complexity and runs fast enough to make time-dependent velocity fields feasible. The evaluation of the integrals leads to increased precision and the unconditional stability. We demonstrate by numerical examples that the ELM with our proposed algorithm for accurate numerical integration has the following two features: first it is much more accurate and more stable than the ones with traditional numerical integration techniques and secondly the overall cost of the proposed method is comparable with the traditional ones.
