An Efficient Fourth Order Implicit Runge-Kutta Algorithm for Second Order Systems

2008 ◽  
pp. 169-178
Author(s):  
Basem S. Attili
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Runge Kutta ◽  
Second Order Systems

Modelling the effects of diffusive-viscous waves in a 3-D fluid-saturated media using two numerical approaches

2020 ◽  
Vol 224 (2) ◽  
pp. 1443-1463
Author(s):  
Victor Mensah ◽  
Arturo Hidalgo
Keyword(s):  
Seismic Wave ◽  
Fourth Order ◽  
Seismic Waves ◽  
Time Discretization ◽  
Seismic Wave Propagation ◽  
Second Order ◽  
Finite Volume Scheme ◽  
Fluid Saturation ◽  
Runge Kutta ◽  
Saturated Medium

SUMMARY The accurate numerical modelling of 3-D seismic wave propagation is essential in understanding details to seismic wavefields which are, observed on regional and global scales on the Earth’s surface. The diffusive-viscous wave (DVW) equation was proposed to study the connection between fluid saturation and frequency dependence of reflections and to characterize the attenuation property of the seismic wave in a fluid-saturated medium. The attenuation of DVW is primarily described by the active attenuation parameters (AAP) in the equation. It is, therefore, imperative to acquire these parameters and to additionally specify the characteristics of the DVW. In this paper, quality factor, Q is used to obtain the AAP, and they are compared to those of the visco-acoustic wave. We further derive the 3-D numerical schemes based on a second order accurate finite-volume scheme with a second order Runge–Kutta approximation for the time discretization and a fourth order accurate finite-difference scheme with a fourth order Runge–Kutta approximation for the time discretization. We then simulate the propagation of seismic waves in a 3-D fluid-saturated medium based on the derived schemes. The numerical results indicate stronger attenuation when compared to the visco-acoustic case.

On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions

2016 ◽  
Vol 26 (3/4) ◽  
pp. 698-721 ◽  
Author(s):  
J I Ramos
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Finite Difference Methods ◽  
Second Order ◽  
Time Step ◽  
Content Type ◽  
Difference Methods ◽  
Temporal And Spatial ◽  
Runge Kutta ◽  
Grlw Equation

Purpose – The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions. Design/methodology/approach – Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained. Findings – It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation. Originality/value – This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

2011 ◽  
Vol 139 (9) ◽  
pp. 2962-2975 ◽  
Author(s):  
William C. Skamarock ◽  
Almut Gassmann
Keyword(s):  
Fourth Order ◽  
Time Integration ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
Diffusion Term ◽  
The Third ◽  
Integration Schemes ◽  
Order Scheme ◽  
Runge Kutta

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

Corners Give Problems When Decoupling Fourth Order Equations Into Second Order Systems

2012 ◽  
Vol 50 (3) ◽  
pp. 1604-1623 ◽  
Author(s):  
Tymofiy Gerasimov ◽  
Athanasios Stylianou ◽  
Guido Sweers
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Second Order Systems ◽  
Fourth Order Equations

An efficient implicit Runge–Kutta method for second order systems

2006 ◽  
Vol 178 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Basem S. Attili ◽  
Khalid Furati ◽  
Muhammed I. Syam
Keyword(s):  
Second Order ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Second Order Systems

scholarly journals Exponentially Fitted and Trigonometrically Fitted Two-Derivative Runge-Kutta-Nyström Methods for Solving y′′(x)=fx,y,y′

2018 ◽  
Vol 2018 ◽  
pp. 1-19
Author(s):  
Tahani Salama Mohamed ◽  
Norazak Senu ◽  
Zarina Bibi Ibrahim ◽  
Nik Mohd Asri Nik Long
Keyword(s):  
Numerical Experiments ◽  
Second Order ◽  
Linear Combinations ◽  
New Approaches ◽  
Nyström Methods ◽  
Exponentially Fitted ◽  
Runge Kutta ◽  
Second Order Systems

Two exponentially fitted and trigonometrically fitted explicit two-derivative Runge-Kutta-Nyström (TDRKN) methods are being constructed. Exponentially fitted and trigonometrically fitted TDRKN methods have the favorable feature that they integrate exactly second-order systems whose solutions are linear combinations of functions {exp⁡(wx),exp⁡(-wx)} and {sin⁡(wx),cos⁡(wx)} respectively, when w∈R, the frequency of the problem. The results of numerical experiments showed that the new approaches are more efficient than existing methods in the literature.

Analysis of different second order systems via runge-kutta method

1999 ◽  
Vol 70 (3) ◽  
pp. 477-493 ◽  
Author(s):  
K. Murugesan ◽  
D.Paul Dhabaran ◽  
David J. Evans
Keyword(s):  
Second Order ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Second Order Systems
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