The Eulerian-Lagrangian Method with Accurate Numerical Integration (Advances in Applied Mathematics and Mechanics, 2018)
This paper is devoted to the study of the Eulerian-Lagrangian method(ELM) for convection-diffusion equations on unstructured grids with or withoutaccurate numerical integration. We first propose an efficient and accurate algorithmto calculate the integrals in the Eulerian-Lagrangian method. Our approachis based on an algorithm for finding the intersection of two non-matching grids. Ithas optimal algorithmic complexity and runs fast enough to make time-dependentvelocity fields feasible. The evaluation of the integrals leads to increased precisionand the unconditional stability. We demonstrate by numerical examples that theELM with our proposed algorithm for accurate numerical integration has the followingtwo features: firstly, it is much more accurate and more stable than the oneswith traditional numerical integration techniques and secondly, the overall cost ofthe proposed method is comparable with the traditional ones.