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Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-<jats:italic>r<\/jats:italic> for <jats:inline-formula><jats:alternatives><jats:tex-math>$$r\\ge 4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>r<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating <jats:italic>p<\/jats:italic>-harmonic flows using our proposed method.\n<\/jats:p>","DOI":"10.1007\/s10915-024-02702-2","type":"journal-article","created":{"date-parts":[[2024,10,26]],"date-time":"2024-10-26T19:02:24Z","timestamp":1729969344000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres"],"prefix":"10.1007","volume":"101","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4549-2761","authenticated-orcid":false,"given":"Shingyu","family":"Leung","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wai Ming","family":"Chau","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Young Kyu","family":"Lee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,10,26]]},"reference":[{"key":"2702_CR1","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1007\/BF01211069","volume":"104","author":"SL Adler","year":"1986","unstructured":"Adler, S.L.: Quaternionic Quantum Field Theory. 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