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Comparing numerical integration schemes for a car-following model with real-world data

Přikryl, Jan, Vaniš, Miroslav (2017)

Programs and Algorithms of Numerical Mathematics

A key element of microscopic traffic flow simulation is the so-called car-following model, describing the way in which a typical driver interacts with other vehicles on the road. This model is typically continuous and traffic micro-simulator updates its vehicle positions by a numerical integration scheme. While increasing the order of the scheme should lead to more accurate results, most micro-simulators employ the simplest Euler rule. In our contribution, inspired by [1], we will provide some additional...

Energy-preserving Runge-Kutta methods

Elena Celledoni, Robert I. McLachlan, David I. McLaren, Brynjulf Owren, G. Reinout W. Quispel, William M. Wright (2009)

ESAIM: Mathematical Modelling and Numerical Analysis

We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.

Euler scheme for SDEs with non-Lipschitz diffusion coefficient : strong convergence

Abdel Berkaoui, Mireille Bossy, Awa Diop (2008)

ESAIM: Probability and Statistics

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form | x | α , α ∈ [ 1 / 2 , 1 ) . In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

Euler scheme for SDEs with non-Lipschitz diffusion coefficient: strong convergence

Abdel Berkaoui, Mireille Bossy, Awa Diop (2007)

ESAIM: Probability and Statistics

We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form |x|α, α ∈ [1/2,1). In that case, we study the rate of convergence of a symmetrized version of the Euler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz.

Explicit two-step Runge-Kutta methods

Zdzisław Jackiewicz, Rosemary Anne Renaut, Marino Zennaro (1995)

Applications of Mathematics

The explicit two-step Runge-Kutta (TSRK) formulas for the numerical solution of ordinary differential equations are analyzed. The order conditions are derived and the construction of such methods based on some simplifying assumptions is described. Order barriers are also presented. It turns out that for order p ≤ 5 the minimal number of stages for explicit TSRK method of order p is equal to the minimal number of stages for explicit Runge-Kutta method of order p - 1 . Numerical results are presented which...

Fully discrete error estimation by the method of lines for a nonlinear parabolic problem

Tomáš Vejchodský (2003)

Applications of Mathematics

A posteriori error estimates for a nonlinear parabolic problem are introduced. A fully discrete scheme is studied. The space discretization is based on a concept of hierarchical finite element basis functions. The time discretization is done using singly implicit Runge-Kutta method (SIRK). The convergence of the effectivity index is proven.

Currently displaying 21 – 40 of 106