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The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell\u2019s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.\n<\/jats:p>","DOI":"10.1007\/s10915-024-02462-z","type":"journal-article","created":{"date-parts":[[2024,3,4]],"date-time":"2024-03-04T11:01:36Z","timestamp":1709550096000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics"],"prefix":"10.1007","volume":"99","author":[{"given":"Erik","family":"Chudzik","sequence":"first","affiliation":[]},{"ORCID":"http:\/\/orcid.org\/0000-0002-9901-7149","authenticated-orcid":false,"given":"Christiane","family":"Helzel","sequence":"additional","affiliation":[]},{"given":"M\u00e1ria","family":"Luk\u00e1\u010dov\u00e1-Medvid\u2019ov\u00e1","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,3,4]]},"reference":[{"issue":"1","key":"2462_CR1","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1007\/s42967-021-00175-w","volume":"5","author":"R Abgrall","year":"2023","unstructured":"Abgrall, R.: A combination of residual distribution and the active flux formulations or a new class of schemes that can combine several writings of the same hyperbolic problem: application to the 1d Euler equations. 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