{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:19:02Z","timestamp":1757618342750,"version":"3.44.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,6,7]],"date-time":"2025-06-07T00:00:00Z","timestamp":1749254400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,7]],"date-time":"2025-06-07T00:00:00Z","timestamp":1749254400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-19-1-0281","FA9550-17-1-0394"],"award-info":[{"award-number":["FA9550-19-1-0281","FA9550-17-1-0394"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1912183"],"award-info":[{"award-number":["DMS-1912183"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023164"],"award-info":[{"award-number":["DE-SC0023164"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In this paper, we introduce a new family of spatially co-located field solvers for particle-in-cell applications which evolve the potential formulation of Maxwell\u2019s equations under the Lorenz gauge. Our recent work [2] introduced the concept of time-consistency, which connects charge conservation to the preservation of the gauge at the semi-discrete level. It will be shown that there exists a large family of time discretizations which satisfy this property. Additionally, it will be further shown that for large classes of time marching methods, the satisfaction of the gauge condition automatically implies the satisfaction of Gauss\u2019s law for electricity, with the potential formulation ensuring that that Gauss\u2019s law for magnetism is satisfied by definition. We focus on popular time marching methods including centered differences, backward differences, and diagonally-implicit Runge-Kutta methods, which are coupled to a spectral discretization in space. We demonstrate the theory by testing the methods on a relativistic Weibel instability and a drifting cloud of electrons.<\/jats:p>","DOI":"10.1007\/s10915-025-02953-7","type":"journal-article","created":{"date-parts":[[2025,6,7]],"date-time":"2025-06-07T01:36:09Z","timestamp":1749260169000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Particle-in-cell Method for Plasmas with A Generalized Momentum Formulation, Part III: A family of Gauge Conserving Methods"],"prefix":"10.1007","volume":"104","author":[{"given":"Andrew J.","family":"Christlieb","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"William A.","family":"Sands","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8308-8110","authenticated-orcid":false,"given":"Stephen R.","family":"White","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,7]]},"reference":[{"key":"2953_CR1","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s10915-025-02824-1","volume":"103","author":"AJ Christlieb","year":"2025","unstructured":"Christlieb, A.J., Sands, W.A., White, S.R.: A particle-in-cell method for plasmas with a generalized momentum formulation, part I: Model formulation. J. Sci. Comput. 103, 15 (2025). https:\/\/doi.org\/10.1007\/s10915-025-02824-1","journal-title":"J. Sci. Comput."},{"key":"2953_CR2","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1007\/s10915-024-02728-6","volume":"101","author":"AJ Christlieb","year":"2024","unstructured":"Christlieb, A.J., Sands, W.A., White, S.R.: A particle-in-cell method for plasmas with a generalized momentum formulation, part II: Enforcing the Lorenz gauge condition. J. Sci. Comput. 101, 73 (2024). https:\/\/doi.org\/10.1007\/s10915-024-02728-6","journal-title":"J. Sci. Comput."},{"issue":"2","key":"2953_CR3","doi-asserted-by":"publisher","first-page":"896","DOI":"10.1007\/s10915-016-0268-8","volume":"70","author":"M Causley","year":"2017","unstructured":"Causley, M., Christlieb, A., Wolf, E.: Method of lines transpose: An efficient unconditionally stable solver for wave propagation. J. Sci. Comput. 70(2), 896\u2013921 (2017)","journal-title":"J. Sci. Comput."},{"key":"2953_CR4","doi-asserted-by":"publisher","DOI":"10.1063\/5.0019210","volume":"10","author":"M Thavappiragasm","year":"2020","unstructured":"Thavappiragasm, M., Christlieb, A.J., Luginsland, J., Guthrey, P.T.: A fast local embedded boundary method suitable for high power electromagnetic sources. AIP Adv. 10, 115318 (2020)","journal-title":"AIP Adv."},{"issue":"4","key":"2953_CR5","doi-asserted-by":"publisher","first-page":"1790","DOI":"10.1109\/TPS.2008.927143","volume":"36","author":"C Benedetti","year":"2008","unstructured":"Benedetti, C., Sgattoni, A., Turchetti, G., Londrillo, P.: $${ aladyn}$$: A high-accuracy pic code for the maxwell-vlasov equations. IEEE Trans. Plasma Sci. 36(4), 1790\u20131798 (2008). https:\/\/doi.org\/10.1109\/TPS.2008.927143","journal-title":"IEEE Trans. Plasma Sci."},{"issue":"1","key":"2953_CR6","doi-asserted-by":"publisher","first-page":"52","DOI":"10.3847\/1538-4357\/aa6d13","volume":"841","author":"M Shalaby","year":"2017","unstructured":"Shalaby, M., Broderick, A.E., Chang, P., Pfrommer, C., Lamberts, A., Puchwein, E.: Sharp: A spatially higher-order, relativistic particle-in-cell code. Astrophys J. 841(1), 52 (2017). https:\/\/doi.org\/10.3847\/1538-4357\/aa6d13","journal-title":"Astrophys J."},{"key":"2953_CR7","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1016\/j.jcp.2014.10.064","volume":"281","author":"A Davidson","year":"2015","unstructured":"Davidson, A., Tableman, A., An, W., Tsung, F.S., Lu, W., Vieira, J., Fonseca, R.A., Silva, L.O., Mori, W.B.: Implementation of a hybrid particle code with a pic description in r-z and a gridless description in $$\\phi $$ into osiris. J. Comput. Phys. 281, 1063\u20131077 (2015). https:\/\/doi.org\/10.1016\/j.jcp.2014.10.064","journal-title":"J. Comput. Phys."},{"key":"2953_CR8","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.enganabound.2023.12.028","volume":"160","author":"A Sadeghirad","year":"2024","unstructured":"Sadeghirad, A.: B-spline convected particle domain interpolation method. Eng. Anal. Boundary Elem. 160, 106\u2013133 (2024). https:\/\/doi.org\/10.1016\/j.enganabound.2023.12.028","journal-title":"Eng. Anal. Boundary Elem."},{"issue":"1","key":"2953_CR9","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1137\/130932685","volume":"52","author":"MF Causley","year":"2014","unstructured":"Causley, M.F., Christlieb, A.J.: Higher order A-stable schemes for the wave equation using a successive convolution approach. SIAM J. Numer. Anal. 52(1), 220\u2013235 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"2953_CR10","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1088\/0741-3335\/47\/5A\/017","volume":"47","author":"JP Verboncoeur","year":"2005","unstructured":"Verboncoeur, J.P.: Particle simulation of plasmas: Review and advances. Plasma Phys. Controlled Fusion 47, 231\u2013260 (2005)","journal-title":"Plasma Phys. Controlled Fusion"},{"key":"2953_CR11","volume-title":"Plasma Physics Via Computer Simulation","author":"CK Birdsall","year":"1985","unstructured":"Birdsall, C.K., Langdon, A.B.: Plasma Physics Via Computer Simulation. McGraw-Hill Book Company, New York, NY (1985)"},{"key":"2953_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6333-3","volume-title":"A Practical Guide to Splines","author":"C deBoor","year":"1978","unstructured":"deBoor, C.: A Practical Guide to Splines. Springer, New York, NY (1978)"},{"key":"2953_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1063\/1.4994705","volume":"24","author":"L Siddi","year":"2017","unstructured":"Siddi, L., Lapenta, G., Gibbon, P.: Mesh-free Hamiltonian implementation of two dimensional Darwin model. Phys. Plasmas 24, 1\u201311 (2017)","journal-title":"Phys. Plasmas"},{"key":"2953_CR14","doi-asserted-by":"crossref","unstructured":"Birdsall, C.K., Langdon, A.B.: Particle simulation techniques. In: Computer Applications in Plasma Science and Engineering, pp. 7\u201341. Springer, New York, NY (1991)","DOI":"10.1007\/978-1-4612-3092-2_2"},{"issue":"2","key":"2953_CR15","doi-asserted-by":"publisher","first-page":"658","DOI":"10.1016\/j.jcp.2006.01.039","volume":"217","author":"C Huang","year":"2006","unstructured":"Huang, C., Decyk, V.K., Ren, C., Zhou, M., Lu, W., Mori, W.B., Cooley, J.H., Antonsen, T.M., Jr., Katsouleas, T.: Quickpic: A highly efficient particle-in-cell code for modeling wakefield acceleration in plasmas. J. Comput. Phys. 217(2), 658\u2013679 (2006)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"2953_CR16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.78.016404","volume":"78","author":"E Cormier-Michel","year":"2008","unstructured":"Cormier-Michel, E., Shadwick, B.A., Geddes, C.G.R., Esarey, E., Schroeder, C.B., Leemans, W.P.: Unphysical kinetic effects in particle-in-cell modeling of laser wakefield accelerators. Phys. Rev. E-Statistical, Nonlinear, and Soft Matter Phys. 78(1), 016404 (2008)","journal-title":"Phys. Rev. E-Statistical, Nonlinear, and Soft Matter Phys."},{"issue":"12","key":"2953_CR17","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevSTAB.17.121301","volume":"17","author":"R Lehe","year":"2014","unstructured":"Lehe, R., Thaury, C., Guillaume, E., Lifschitz, A., Malka, V.: Laser-plasma lens for laser-wakefield accelerators. Phys. Rev. Spec. Topics-Accelerators and Beams 17(12), 121301 (2014)","journal-title":"Phys. Rev. Spec. Topics-Accelerators and Beams"},{"key":"2953_CR18","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1016\/0010-4655(92)90169-Y","volume":"69","author":"J Villasenor","year":"1992","unstructured":"Villasenor, J., Buneman, O.: Rigorous charge conservation for local electromagnetic field solvers. Comput. Phys. Commun. 69, 306\u2013316 (1992)","journal-title":"Comput. Phys. Commun."},{"key":"2953_CR19","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1103\/PhysRevLett.2.83","volume":"2","author":"ES Weibel","year":"1959","unstructured":"Weibel, E.S.: Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution. Phys. Rev. Lett. 2, 83\u201384 (1959)","journal-title":"Phys. Rev. Lett."},{"key":"2953_CR20","first-page":"205","volume":"10","author":"Q Meng-zhao","year":"1992","unstructured":"Meng-zhao, Q., Mei-qing, Z.: Symplectic runge-kutta algorithms for hamiltonian systems. J. Comput. Math. 10, 205\u2013215 (1992)","journal-title":"J. Comput. Math."},{"key":"2953_CR21","unstructured":"Kennedy, C., Carpenter, M.: Diagonally implicit runge-kutta methods for ordinary differential equations. a review. Technical Report 20160005923, NASA (March 2016)"},{"key":"2953_CR22","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.126.095101","volume":"126","author":"A Bohdan","year":"2021","unstructured":"Bohdan, A., Pohl, M., Niemiec, J., Morris, P.J., Matsumoto, Y., Amano, T., Hoshino, M., Sulaiman, A.: Magnetic field amplification by the weibel instability at planetary and astrophysical shocks with high mach number. Phys. Rev. Lett. 126, 095101 (2021). https:\/\/doi.org\/10.1103\/PhysRevLett.126.095101","journal-title":"Phys. Rev. Lett."},{"issue":"1","key":"2953_CR23","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1016\/j.cpc.2005.03.036","volume":"169","author":"S Atzeni","year":"2005","unstructured":"Atzeni, S., Schiavi, A., Califano, F., Cattani, F., Cornolti, F., Del Sarto, D., Liseykina, T.V., Macchi, A., Pegoraro, F.: Fluid and kinetic simulation of inertial confinement fusion plasmas. Comput. Phys. Commun. 169(1), 153\u2013159 (2005). https:\/\/doi.org\/10.1016\/j.cpc.2005.03.036","journal-title":"Comput. Phys. Commun."},{"issue":"5","key":"2953_CR24","doi-asserted-by":"publisher","first-page":"1979","DOI":"10.1063\/1.1556605","volume":"10","author":"RA Fonseca","year":"2003","unstructured":"Fonseca, R.A., Silva, L.O., Tonge, J.W., Mori, W.B., Dawson, J.M.: Three-dimensional Weibel instability in astrophysical scenarios. Phys. Plasmas 10(5), 1979\u20131984 (2003)","journal-title":"Phys. Plasmas"},{"key":"2953_CR25","doi-asserted-by":"publisher","first-page":"830","DOI":"10.1063\/1.1693518","volume":"14","author":"RL Morse","year":"1971","unstructured":"Morse, R.L., Nielson, C.W.: Numerical simulation of the weibel instability in one and two dimensions. Phys. Fluids 14, 830\u2013840 (1971)","journal-title":"Phys. Fluids"},{"issue":"10","key":"2953_CR26","doi-asserted-by":"publisher","first-page":"2391","DOI":"10.1016\/j.cpc.2014.05.010","volume":"185","author":"G Chen","year":"2014","unstructured":"Chen, G., Chac\u00f3n, L.: An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1d\u20133v Vlasov-Darwin particle-in-cell algorithm. Comput. Phys. Commun. 185(10), 2391\u20132402 (2014)","journal-title":"Comput. Phys. Commun."},{"issue":"3","key":"2953_CR27","doi-asserted-by":"publisher","first-page":"1390","DOI":"10.1103\/PhysRevLett.31.1390","volume":"31","author":"R Lee","year":"1973","unstructured":"Lee, R., Lampe, M.: Electromagnetic instabilities, filamentation, and focusing of relativistic electron beams. Phys. Rev. Lett. 31(3), 1390\u20131393 (1973). https:\/\/doi.org\/10.1103\/PhysRevLett.31.1390","journal-title":"Phys. Rev. Lett."},{"key":"2953_CR28","unstructured":"Christlieb, A., Gong, S., Yang, H.: Boundary corrections for kernel approximation to differential operators (2024). arXiv:2410.09332"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02953-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-025-02953-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-025-02953-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T18:42:16Z","timestamp":1757184136000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-025-02953-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,7]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["2953"],"URL":"https:\/\/doi.org\/10.1007\/s10915-025-02953-7","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2025,6,7]]},"assertion":[{"value":"25 October 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 June 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"38"}}