On the implementation of low-dissipative Runge–Kutta projection methods for time dependent flows using OpenFOAM®

2014 ◽  
Vol 93 ◽  
pp. 153-163 ◽  
Author(s):  
V. Vuorinen ◽  
J.-P. Keskinen ◽  
C. Duwig ◽  
B.J. Boersma
Keyword(s):  
Projection Methods ◽  
Time Dependent ◽  
Runge Kutta

Modeling on Falling Velocity of Sodiumtetraborate Aqueous Solution Drops before the Gelation of PVA-TiO2 Suspensions by the Runge-Kutta Method in Matlab 6.5

2008 ◽  
Vol 368-372 ◽  
pp. 1683-1685
Author(s):  
Cheng Long Yu ◽  
Xiu Feng Wang ◽  
Jun Xin Zhou ◽  
Hong Tao Jiang ◽  
Yan Wang
Keyword(s):  
Differential Equation ◽  
Aqueous Solution ◽  
Ordinary Differential Equation ◽  
Elevation Angle ◽  
Time Dependent ◽  
Quadratic Term ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Air Resistance ◽  
Runge Kutta

Numerical modeling on falling of sodiumtetraborate aqueous solution drops as the initiator before the gelation of PVA-TiO2 suspensions was conducted. Effect of time and elevation angle of the PVA-TiO2 suspensions on the falling velocity of the sodiumtetraborate aqueous solution drops was analyzed. An ordinary differential equation was given. Integration of the ordinary differential equation was fulfilled using the fourth-order Runge-Kutta method in Matlab 6.5. From the model, a two-order nonlinear effect of time on the velocity of the drops during falling is determined and the quadratic term -3.408t2 serves as the time dependent air resistance. The component of the falling velocity along the suspensions increases with the increasing of the elevation angle. However, for the component vertical to the suspensions, with elevation angle increasing, it decreases.

scholarly journals Runge–Kutta convolution coercivity and its use for time-dependent boundary integral equations

2018 ◽  
Vol 39 (3) ◽  
pp. 1134-1157
Author(s):  
Lehel Banjai ◽  
Christian Lubich
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Scattering Problem ◽  
Time Discretization ◽  
Boundary Integral ◽  
Time Dependent ◽  
Stability And Convergence ◽  
Linear Wave ◽  
Convolution Quadrature ◽  
Runge Kutta

Abstract A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretizations. It is known that this coercivity property is inherited by convolution quadrature time discretization based on A-stable multistep methods, which are of order at most 2. Here we study the question as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge–Kutta methods and hence for methods of arbitrary order. As an illustration the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretization of a nonlinear boundary integral equation that originates from a nonlinear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretization.

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