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The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.&lt;\/p&gt;&lt;\/abstract&gt;<\/jats:p>","DOI":"10.3934\/nhm.2023017","type":"journal-article","created":{"date-parts":[[2023,1,9]],"date-time":"2023-01-09T11:34:11Z","timestamp":1673264051000},"page":"412-443","source":"Crossref","is-referenced-by-count":2,"title":["Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays"],"prefix":"10.3934","volume":"18","author":[{"given":"Dingwen","family":"Deng","sequence":"first","affiliation":[]},{"given":"Jingliang","family":"Chen","sequence":"additional","affiliation":[]}],"member":"2321","reference":[{"key":"key-10.3934\/nhm.2023017-1","doi-asserted-by":"publisher","unstructured":"S. A. Messaoudi, A. Fareh, N. 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