We consider numerical techniques to solve stiff initial value problems (IVPs) given by
$$\begin{array}{}{\displaystyle \frac{dy}{dt}=f(t,y(t)),\phantom{\rule{2em}{0ex}}t\in [{t}_{0},{t}_{f}],\phantom{\rule{2em}{0ex}}y({t}_{0})={y}_{0},}\end{array}$$(1)

where *f* has continuously bounded partial derivatives up to required order for the developed numerical method. The stiff systems are broadly classified into two categories - one is to have stiff components in just a given system and the other is to have the components in both the system and its solution. In the first case, solutions of the system behaves smoothly as time is increasing so it can be easily solved by any implicit scheme with an appropriate step size. On the other hand, the solutions of the given system in the second case have stiff components expressed as irregularities or sharp fronts in some or whole time intervals. In the intervals, we carefully handle a numerical scheme since the solutions are very rapidly changed. Most interesting research topics, induced from the real applications such as fluid dynamics, molecular dynamics, plasma or other physics, are related with the second case.

There are lots of numerical strategies to find efficient and accurate solutions of the stiff systems. In this paper, we focus on the parallelization scheme to find the efficient solutions of the stiff IVPs. Time parallelization scheme has received a lot of attention over the past few years and several parallelization schemes have been proposed [1, 2, 3]. Especially in 2001, a new algorithm which was named parareal algorithm for the solution of time dependent differential equations in parallel was introduced [4]. It can be defined by
$$\begin{array}{}{y}_{n+1}^{k}=G({t}_{n+1},{t}_{n},{y}_{n}^{k})+F({t}_{n+1},{t}_{n},{y}_{n}^{k-1})-G({t}_{n+1},{t}_{n},{y}_{n}^{k-1})\end{array}$$(2)

where the subscript *n* refers to the time subdomain number, the superscript *k* refers to the iteration number. *F* represents a fine propagator, that is, a more accurate solution on a fine grid in time interval [*t*_{n}, *t*_{n+1}] with an initial value
$\begin{array}{}{y}_{n}^{k-1}\end{array}$ *G* represents a coarse propagator, a less accurate approximation in a coarser grid. Note that the F-propagator determines the overall accuracy of the parareal method, whereas the convergence order of the method is decided by the order of the G-propagator and the number of iterations used when it is coupled with a sufficiently accurate F-propagator [5, 6]. Unfortunately, the traditional parareal scheme has the low parallel efficiency which is bounded by 1/K, where K is the parareal iterations needed to converge to the desired accuracy. In most case, 2 or more iterations are needed, so the efficiency of the traditional parareal scheme is less than 50 percents and even worst in practice. To hurdle this drawback, several advanced parareal techniques based on the deferred correction (DC) methods have been recently introduced [2, 5, 6], in which DC strategies are utilized within the parareal iteration for the F-propagator by using one or few DC iterations during each parareal iteration.

In the parareal algorithm, there is an important assumption that there are an infinite number of processors to use, so each processor is assigned to each different time interval with a uniform time step size. However, only a few finite number of processors can be provided in practice. Even if an infinite number of processors are provided, it is not efficient to assign the uniform step size on each processor without any consideration on the property of the problem, especially for stiff systems or partial differential equations (PDEs) having sharp front. That is, it is more efficient that a larger time step size is assigned for smooth or non-stiff regions in solutions, while a smaller time step size is needed for shock or stiff regions. Therefore, the usage of adaptive step sizes is very important issue to improve the efficiency of the parareal algorithm. Related to this issue, many researchers have attempted to find a suitable way to automatically detect stiffness [7].

The aims of this paper are to introduce a criterion to detect stiffness and to develop a scheme for finding an adaptive time step size according to the extent of stiffness in each interval. First of all, for the given system, we need to split the stiff and non-stiff parts in a given time domain. There are various ways to detect stiffness. For simplicity’s sake, we examine a gradient ratio of a given system to split the stiff and non-stiff parts. Once the stiff regions is detected, the corresponding step size should be automatically controlled. So, the time intervals in stiff regions should be gradually shrank depending on the extent of the stiffness, while those in non-stiff regions are comparatively stretched. Based on these processes, an appropriate time step size for the parareal algorithm is chosen in the sequential step depending on the stiffness at each time interval. Note that in the traditional parareal algorithm, a G-propagator approximates initial values for all time intervals sequentially, with having a uniform step size at the initialization step.

Additionally, a theoretical analysis of the parareal algorithm shows that the stability of the method depends on the choice of G-propagator [5, 8]. Especially, for solving highly stiff problem, G-propagator should be satisfied an L-stability. Also, each time interval is determined in the initialization step with a G-propagator, the computational cost for G-propagator should be small enough. Overall, Backward Euler (BE) method will be a good candidate for the G-propagator, since BE is unconditionally stable, its computational costs is relatively small and it has L-stability [9], where it can unconditionally fulfill the stability condition of the parareal methods with less computational costs.

This paper is organized as follows. In Sec. 2, we briefly describe the original parareal technique and the improved parareal algorithms based on the original one. In Sec. 3, we introduce several parameters for detecting degree of stiffness in each interval and discuss a strategy to select adaptive time step size using stiffness detection to improve the overall efficiency of the parareal algorithm. In Sec. 4, preliminary numerical results are presented to show the efficiency of the proposed scheme. Finally in Sec. 5, future research directions are provided.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.