  In this paper we propose an adaptive scheme for the solution of
time-dependent boundary value problems (BVPs). To solve numerically these
problems, we consider the kernel-based method of lines that allows us to split
the spatial and time derivatives, dealing with each separately. This adaptive
algorithm is based on a leave-one-out cross validation (LOOCV) technique, which
is employed as an error indicator. By this scheme, we can first detect the
domain areas where the error is estimated to be too large -- generally due to
steep variations or quick changes in the solution -- and then accordingly
enhance the numerical solution by applying a two-point refinement strategy.
Numerical experiments show the efficacy and performance of our adaptive
refinement method.
