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These methods aim to provide a numerical approximation of the solution by computing its inverse Laplace transform. The choice of the integration contour is determined by the computation of a few suitably weighted pseudo-spectral level sets of the leading operator of the equation. Parabolic and hyperbolic profiles proposed in the literature are investigated and compared to the elliptic contour originally proposed by Guglielmi, L\u00f3pez-Fern\u00e1ndez and Nino 2020, see Guglielmi et al. (Math Comput 89:1161\u20131191, 2020). In summary, the article <jats:list list-type=\"order\">\n                \n                  \n                  <jats:list-item>\n                    <jats:p>provides a comparison among three different integration profiles;<\/jats:p>\n                  <\/jats:list-item>\n                \n                \n                  \n                  <jats:list-item>\n                    <jats:p>proposes a new fast pseudospectral roaming method;<\/jats:p>\n                  <\/jats:list-item>\n                \n                \n                  \n                  <jats:list-item>\n                    <jats:p>optimizes the selection of time windows on which one may arbitrarily approximate the solution by no extra computational cost with respect to the case of a fixed time instant;<\/jats:p>\n                  <\/jats:list-item>\n                \n                \n                  \n                  <jats:list-item>\n                    <jats:p>focuses extensively on computational aspects and it is the reference of the MATLAB code [20], where all algorithms described here are implemented.<\/jats:p>\n                  <\/jats:list-item>\n                \n              <\/jats:list><\/jats:p>","DOI":"10.1007\/s10915-021-01601-0","type":"journal-article","created":{"date-parts":[[2021,9,7]],"date-time":"2021-09-07T04:02:50Z","timestamp":1630987370000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Pseudospectral Roaming Contour Integral Methods for Convection-Diffusion Equations"],"prefix":"10.1007","volume":"89","author":[{"given":"Nicola","family":"Guglielmi","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3951-7296","authenticated-orcid":false,"given":"Mar\u00eda","family":"L\u00f3pez-Fern\u00e1ndez","sequence":"additional","affiliation":[]},{"given":"Mattia","family":"Manucci","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,9,7]]},"reference":[{"issue":"2","key":"1601_CR1","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1137\/100788860","volume":"33","author":"AH Al-Mohy","year":"2011","unstructured":"Al-Mohy, A.H., Higham, N.J.: Computing the action of the matrix exponential, with an application to exponential integrators. 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