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Comput."],"published-print":{"date-parts":[[2024,2,29]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We develop two new sets of stable, rank-adaptive dynamically orthogonal Runge\u2013Kutta (DORK) schemes that capture the high-order curvature of the nonlinear low-rank manifold. The DORK schemes asymptotically approximate the truncated singular value decomposition at a greatly reduced cost while preserving mode continuity using newly derived retractions. We show that arbitrarily high-order optimal perturbative retractions can be obtained, and we prove that these new retractions are stable. In addition, we demonstrate that repeatedly applying retractions yields a gradient-descent algorithm on the low-rank manifold that converges superlinearly when approximating a low-rank matrix. When approximating a higher-rank matrix, iterations converge linearly to the best low-rank approximation. We then develop a rank-adaptive retraction that is robust to overapproximation. Building off of these retractions, we derive two rank-adaptive integration schemes that dynamically update the subspace upon which the system dynamics are projected within each time step: the stable, optimal dynamically orthogonal Runge\u2013Kutta (so-DORK) and gradient-descent dynamically orthogonal Runge\u2013Kutta (gd-DORK) schemes. These integration schemes are numerically evaluated and compared on an ill-conditioned matrix differential equation, an advection-diffusion partial differential equation, and a nonlinear, stochastic reaction-diffusion partial differential equation. Results show a reduced error accumulation rate with the new stable, optimal and gradient-descent integrators. In addition, we find that rank adaptation allows for highly accurate solutions while preserving computational efficiency.<\/jats:p>","DOI":"10.1137\/22m1534948","type":"journal-article","created":{"date-parts":[[2024,2,8]],"date-time":"2024-02-08T04:02:15Z","timestamp":1707364935000},"page":"A529-A560","source":"Crossref","is-referenced-by-count":7,"title":["Stable Rank-Adaptive Dynamically Orthogonal Runge\u2013Kutta Schemes"],"prefix":"10.1137","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8421-3027","authenticated-orcid":true,"given":"Aaron","family":"Charous","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Center for Computational Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1869-3883","authenticated-orcid":true,"given":"Pierre F. 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