We develop two new sets of stable, rank-adaptive Dynamically Orthogonal Runge-Kutta (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-Kutta (so-DORK) and gradient-descent Dynamically Orthogonal Runge-Kutta (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.

翻译：我们开发了两组新的稳定、自适应秩的动态正交龙格-库塔方案，该方案能够捕捉非线性低秩流形的高阶曲率。这些方案通过新推导的回缩，在显著降低计算成本的同时渐进逼近截断奇异值分解，并保持模态连续性。我们证明了可获得任意高阶最优微扰回缩，且证明了这些新回缩是稳定的。此外，我们展示了重复应用回缩会产生低秩流形上的梯度下降算法，该算法在逼近低秩矩阵时超线性收敛。当逼近高秩矩阵时，迭代线性收敛于最佳低秩逼近。随后我们开发了一种对过度逼近具有鲁棒性的自适应秩回缩。基于这些回缩，我们推导出两种自适应秩积分方案，可在每个时间步内动态更新系统动力学所投影的子空间：即稳定的最优动态正交龙格-库塔方案与梯度下降动态正交龙格-库塔方案。这些积分方案在病态矩阵微分方程、对流扩散偏微分方程以及非线性随机反应扩散偏微分方程上进行了数值评估与比较。结果表明，采用新的稳定、最优及梯度下降积分器可降低误差累积速率。此外，我们发现秩自适应在保持计算效率的同时能实现高精度解。