Digital simulation of thermal reactions

Numerical Simulation of the Frank-Kamenetskii PDE: GPU vs. CPU Computing

“…This chapter will not consider the hopscotch scheme directly as an appropriate method for the solution of the Frank-Kamenetskii partial differential equation due to work done by Feldberg [15] which indicated that for large values of β = △t △x 2 the algorithm produces the problem of propagational inadequacy which leads to inaccuracies -similar results were obtained in [22]. Given the improved accuracy of the Crank-Nicolson method incorporating the Newton method [22] -the order of the error for this method is O(△t 2 ) which is only approximately the case for the Crank-Nicolson method without the Newton iteration incorporated [9] -it seems more fitting to consider an improvement in the computing time of this method. Hence a consideration of such an improvement on the algorithm's current running time will be the focus of this chapter.…”

“…The nonlinear source term was kept explicit when the Crank-Nicolson method was employed, as commented on by Britz et al [9] in whose work the nonlinear term was incorporated in an implicit manner in a style more consistent with the Crank-Nicolson method. Britz et al [9] implemented the Crank-Nicolson scheme with the Newton iteration and showed that it outperformed the explicit implementation of the nonlinearity as in [21] in terms of accuracy. However it does require more computer time as would be expected.…”

“…It has been noted by Britz et al [9] that using (12) turns out to be more convenient and accurate. Due to the fact that the point x 0 = 0 would lead to a singularity in equation (1) we structure the code to account for two instances: x = 0andx = 0.…”

“…The methodology will be explained briefly here; the reader is referred to [7][8][9] for clarification.…”

“…In recent work (see [22]) the Crank-Nicolson method was implemented with the Newton iteration as done by Britz et al [9] by computing a correction set in each iteration to obtain approximate values of the dependent variable at the next time step. The efficiency of the Crank-Nicolson scheme, hopscotch scheme (both of these methods were implemented with an explicit and then an implicit discretisation of the source term) and two versions of the Rosenbrock method were compared [22].…”