Positive Second Order Finite Difference Methods on Fokker-Planck Equations with Dirac Initial Data - Application in Finance

2015 ◽  
Author(s):  
Fabien Le Floc'h
Keyword(s):  
Finite Difference ◽  
Initial Data ◽  
Finite Difference Methods ◽  
Second Order ◽  
Data Application ◽  
Difference Methods ◽  
Fokker Planck Equations ◽  
Fokker Planck

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.

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