By Thi Tam Dang
Partial differential equations play an important role in mathematically oriented scientific fields such as physics and engineering. In this work, we are interested in semilinear parabolic partial differential equations, namely diffusion-reaction equations. Diffusion-reaction equations are mathematical models that describe the changes in space and time of the concentration of one or more chemical substances: local chemical reactions, in which the substances are converted into each other, and diffusion, which causes the substances to spread over a surface in space. There are many applications of diffusion reaction equations in chemistry, biology, and physics. The general form of diffusion-reaction problems can be represented as follows
In this problem we impose with nontrivial boundary conditions denoted by B. The boundary usually represents the walls of a container. These walls are impermeable to the substances, so we need a condition to ensure that the substances can’t leak through these walls. The intial data u0 is given.
We want to look for the numerical solution of the diffusion-reaction problems. Splitting methods are well-known method for time integration of partial differential equations. It is used to split the differential operator into several partial flows, where each partial flows can be solved separately in an efficient way or even exactly. We have applied the splitting methods to the diffusion-reaction problems. For more details, we proceed splitting methods directly to the diffusion-reaction problem as follows
The main advantage of splitting methods is the fact that the splitted equations can usually be computed more efficiently compared to applying a numerical method to the full problem. Moreover, splitting methods have better geometric properties.
However, splitting methods face some challenges when applied to such a class of parabolic problems in the presence of nontrivial boundary conditions. In particular, it causes order re- duction for the Strang splitting method. Therefore, splitting methods has to be modified to avoid such an order reduction. In order to avoid the order reduction, we have proposed an alternate splitting which does not suffer from order reduction and thus significantly increases the efficiency of the numerical method. Numerical experiments will be performed for a variety of configurations in both a single dimension and two space dimensions. We also provided a convergence analysis that confirms and explains the behavior observed in the numerical simu- lations.