Volume 4, Issue 2
The Eulerian-Lagrangian Method with Accurate Numerical Integration

Adv. Appl. Math. Mech., 4 (2012), pp. 156-174.

Published online: 2012-04

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• Abstract

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.

• Keywords

Eulerian-Lagrangian method intersection of non-matching grids exact integration

65D18 65D30 65M25 65N30

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@Article{AAMM-4-156, author = {Kun Li}, title = {The Eulerian-Lagrangian Method with Accurate Numerical Integration}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {2}, pages = {156--174}, abstract = {

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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.11-m1192}, url = {http://global-sci.org/intro/article_detail/aamm/112.html} }
TY - JOUR T1 - The Eulerian-Lagrangian Method with Accurate Numerical Integration AU - Kun Li JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 156 EP - 174 PY - 2012 DA - 2012/04 SN - 4 DO - http://dor.org/10.4208/aamm.11-m1192 UR - https://global-sci.org/intro/aamm/112.html KW - Eulerian-Lagrangian method KW - intersection of non-matching grids KW - exact integration AB -

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.

Kun Li. (1970). The Eulerian-Lagrangian Method with Accurate Numerical Integration. Advances in Applied Mathematics and Mechanics. 4 (2). 156-174. doi:10.4208/aamm.11-m1192
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