  The extended Fisher--Kolmogorov (EFK) equation has been used to describe some
phenomena in physical, material and biology systems. In this paper, we propose
a full-rank splitting scheme and a rank-adaptive splitting approach for this
equation. We first use a finite difference method to approximate the space
derivatives. Then, the resulting semi-discrete system is split into two stiff
linear parts and a nonstiff nonlinear part. This leads to our full-rank
splitting scheme. The convergence and the maximum principle of the proposed
scheme are proved rigorously. Based on the frame of the full-rank splitting
scheme, a rank-adaptive splitting approach for obtaining a low-rank solution of
the EFK equation. Numerical examples show that our methods are robust and
accurate. They can also preserve energy dissipation and the discrete maximum
principle.
