{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:25:18Z","timestamp":1787333118948,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline weighted essentially non-oscillatory (WENO) reconstruction with high order flux evaluation. Polyharmonic splines are utilized in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible, and optimal in the associated Sobolev (Beppo-Levi) space. The flux evaluation is accomplished by solving generalized Riemann problems across cell interfaces. The mesh adaptation is performed through an a posteriori error indicator, which relies on the polyharmonic spline reconstruction scheme. The performance of the proposed method is illustrated by a series of numerical experiments, including linear advection, Burgers's equation, Smolarkiewicz's deformational flow test, and the five-spot problem.<\/jats:p>","DOI":"10.1137\/100792573","type":"journal-article","created":{"date-parts":[[2010,11,9]],"date-time":"2010-11-09T18:51:20Z","timestamp":1289328680000},"page":"3251-3277","source":"Crossref","is-referenced-by-count":49,"title":["Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction"],"prefix":"10.1137","volume":"32","author":[{"given":"Terhemen","family":"Aboiyar","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Emmanuil H.","family":"Georgoulis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Armin","family":"Iske","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,9]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1994.1148"},{"key":"R2","unstructured":"T. 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