Numerical Methods for Ordinary Differential Equations: Butcher/Numerical Methods

By Branislav @. Nikoli. 2005

1,993 citations recorded by the source corpus.

Subfields: Numerical Analysis, Optimization & Training, Numerical Methods & Scientific Computing

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Numerical Analysis

1. Jesús Vigo-Aguiar
2. Higinio Ramos
3. V. Chelyshkov
4. Maizar Raissi
5. Chris Lovett
6. G. Em Karniadakis
7. Debadeepta Dey
8. Jasper A. Vrugt

In Optimization & Training

1. G. Em Karniadakis
2. V. Chelyshkov
3. Tong Xiao
4. Jesús Vigo-Aguiar
5. Maizar Raissi
6. Chris Lovett
7. Jingbo Zhu
8. Derek F. Wong

In Numerical Methods & Scientific Computing

1. Maizar Raissi
2. G. Em Karniadakis
3. Paris Perdikaris
4. V. Chelyshkov
5. Jesús Vigo-Aguiar
6. Chris Lovett
7. Debadeepta Dey
8. Jasper A. Vrugt

Citations in the corpus

• Learning Deep Transformer Models for Machine Translation (2019, background): “A possible reason is that only one previous step is used to predict the current value (Butcher, 2003).”
• Denoising Diffusion Implicit Models (2020, methodology): “Moreover, since the sampling procedure of DDIMs is similar to that of an neural ODE, it would be interesting to see if methods that decrease the discretization error in ODEs, including multi-step methods such as Adams-Bashforth (Butcher & Goodwin, 2008), could be helpful for further improving sample”
• AirSim: High-Fidelity Visual and Physical Simulation for Autonomous Vehicles (2017, methodology): “In our implementation we use Verlet integration instead of Runge Kutta for stability and simplicity while being nearly as efficient as Euler integration (see [11] for details).”
• Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations (2017, methodology): “Let us start with the most general form of the linear multistep methods [7] applied to (1), i.”
• Finite difference methods for ordinary and partial differential equations - steady-state and time-dependent problems (2007, background): “Butcher [12] and Hairer, Nørsett, and Wanner [43, 44] provide more recent surveys of the field.”
• State Transitions and the Continuum Limit for a 2D Interacting, Self-Propelled Particle System (2006, background): “One example is a coherent flock formation involving a polarized group moving in the same direction.”
• A DISCUSSION ON EULER METHOD: A REVIEW (2013, neutral): “The advanced numerical methods are themselves complicated but do yield results which cannot be obtained by other simple methods ([6],[7]).”
• High-order linear multistep methods with general monotonicity and boundedness properties (2005, background): “Here, the order conditions give p + 1 relations for the coefficients aj,bj [5,11].”
• Alternative orthogonal polynomials and quadratures. (2006, methodology): “15) is both A- and L-stable (3-stage algorithm Radau IIA, or Ehle method, [1, 6]).”
• High-order quantum algorithm for solving linear differential equations (2010, background): “Another drawback is that this approach does not enable simulation of inhomogeneous differential equations.”