Solving Ordinary Differential Equations I: Nonstiff Problems

2009

664 citations recorded by the source corpus.

Citations in the corpus

• Neural Ordinary Differential Equations (2018, neutral): “There have since been more than 120 years of development of ef fi cient and accurate ODE solvers (Runge, 1895; Kutta, 1901; Hairer et al., 1987).”
• Finding Density Functionals with Machine Learning (2011, neutral): “In Table II”
• A brief introduction to pseudospectral methods : application to diffusion problems (2016, unknown): “As general fundamental references on this topic we can recommend [16, 17].”
• Probabilistic numerics and uncertainty in computations (2015, unknown): “x~dt(t)= f(x(t);t), where x∶R+_RNis a real-valued curve parametrised by tknown to start at the initial value x(t0)=x0. Explicit Runge-Kutta methods are a basic and well-studied tool for such problems [10]—while not necessarily state of the art, they nevertheless perform well on many problems, and ar”
• Numerical Methods for Evolutionary Differential Equations (2008, unknown): “See [82, 41, 150] for more methods and justification of the above claims.”
• Probabilistic ODE Solvers with Runge-Kutta Means (2014, unknown): “As recently pointed out by Hennig and Hauberg [6], since Runge-Kutta methods are linear extrapolation methods, their structure can be emulated by Gaussian process (GP) regression algorithms.”
• Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impacts (2012, unknown): “(61) This practical estimation of the error recalls the oldest device of Runge cited by [20, page 164].”
• Jump-adapted discretization schemes for Lévy-driven SDEs (2010, unknown): “We refer the reader to [11] for details on Runge-Kutta schemes and the corresponding error estimates.”
• Control: A perspective (2014, unknown): “Numerical analysts developed reliable computer code for solving Lyapunov and Riccati equations [415], and for integrating differential and differential-algebraic equations [244,277,276,278,24,274].”
• Optimal tuning of the hybrid Monte Carlo algorithm (2010, unknown): “Because the leapfrog scheme is stable, standard numerical analysis techniques [20] show that it is possible to estimate the global errors in terms of the local residuals (truncation errors).”
• The Magnus expansion and some of its applications (2008, unknown): “Other implicit RK schemes widely used in the literature involving the minimum number of stages at each order are based on Gauss–Legendre collocation points [100].”