Numerical Solution of Ordinary Differential Equations

By V. Ryaben'kii, S. Tsynkov. 2006

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In Numerical Analysis

Citations in the corpus

DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps (2022, background): “We note that for semi-linear ODEs, the solution at time t can be exactly formulated by the “variation of constants” formula [30]: This formulation decouples the linear part and the nonlinear part.”
Applied Numerical Methods with MATLAB for Engineers and Scientists (2004, unknown): “The ode23 function uses the BS23 algorithm (Bogacki and Shampine, 1989; Shampine, 1994), which simultaneously uses second- and third-order RK formulas to solve the ODE and make error estimates for step-size adjustment.”
The Evaluation of American Option Prices Under Stochastic Volatility and Jump-Diffusion Dynamics Using the Method of Lines (2009, unknown): “4All ODEs have been solved by use of the implicit trapezoidal rule (see [31]).”
Modeling and simulation for automatic control (2002, unknown): “There is a large literature on numerical integrators (Hairer, Nørsett and Wanner 1993), (Hairer and Wanner 1996), (Lambert 1991), (Shampine 1994), and a wide range of methods are available.”
OpenSMOKE++: An object-oriented framework for the numerical modeling of reactive systems with detailed kinetic mechanisms (2015, unknown): “The ODE system (19) is usually stiff [45], since the evolution of chemical species in most cases occurs on a wide range of characteristic times.”
Numerical Methods for Evolutionary Differential Equations (2008, background): “7 The following ODE system models a chemical reaction system and has been used extensively as a test problem for stiff solvers [154, 85, 14]: y ′ 1 = −αy1 + βy2y3 , y ′ 2 = αy1 − βy2y3 − γy(2) 2 , y ′ 3 = γy(2) 2 .”
Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. (2005, methodology): “An essential component in these methods is event location in ODEs, namely, methods which solve an ODE for a certain variable f(t) and determine the first time that f(t) crosses zero [34,35].”
Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review (2016, unknown): “Combining (81) with an appropriate starting guess, a modified Newton iteration [180, 210, 371, 372, 378] provides Ui and Fi.”
Probabilistic ODE Solvers with Runge-Kutta Means (2014, background): “From the second interval onward, the RK step solves an estimated IVP, and begins to accumulate a global estimation error not bounded by the convergence order (an effect termed “Lady Windermere’s fan” by Wanner [16]).”
RKC time-stepping for advection-diffusion-reaction problems (2004, methodology): “Variable step sizes are based on a local error per step criterion [14].”
Solving Ordinary Differential Equations in R (2012, background): “An (overly) simple two-compartment model, representing drug concentration in the gut ( y 1 ) and in the blood ( y 2 ) can represent this process [24]: Here a is the absorption rate, b is the removal rate from the blood, and the term u ( t ) represents the daily delivery of the drug to the…”
Numerical Solution of Ordinary Differential Equations: Atkinson/Numerical (2009, methodology): “Also, see Shampine [72] for a thorough study of one-step and multistep m thods and of their implementation in computer software.”
Ancient numerical daemons of conceptual hydrological modeling: 2. Impact of time stepping schemes on model analysis and prediction (2010, unknown): “The reader is also referred to classic texts on numerical ODE methods for further background material [e.g., Lambert, 1991; Shampine, 1994 ].”
A multirate time stepping strategy for stiff ordinary differential equations (2007, unknown): “This form of variable step size selection is standard; see for example [8, 14].”