Adaptive RBF-FD and RBF-HFD schemes for nonlinear parabolic PDE problems in one and two dimensions

Solutions of parabolic PDE problems often exhibit transient behavior from boundary layers or sharp gradients, making accurate and efficient numerical treatment challenging. While uniform discretization with fine refinement can capture these features, it substantially raises computational cost. This work adapts residual subsampling to cluster nodes around transient regions. In one dimension, subsampling uses midpoints of existing nodes, while in two dimensions, sample points arise from quadtree decompositions. The result is a non-uniform discretization with coarser and finer node distributions as needed. Using symbolic computation, we derive radial basis function generated finite difference (RBF-FD) and Hermite finite difference (RBF-HFD) formulas, obtaining analytical expressions for formula weights and local truncation errors. Expressed in terms of shape parameter and inter-nodal distances, these approximations better address stagnation error. The resulting non-uniform grid formulas semi-discretize unsteady PDE problems, with a fifth-order Diagonally Implicit Runge–Kutta (DIRK5) scheme for time marching. At each step, an optimal shape parameter is found by minimize the RMS of truncation errors over adaptive nodes set. We establish consistency, stability, and convergence, then validate performance on benchmark nonlinear parabolic PDEs. Compared to uniform-grid RBF-FD/HFD and global methods, our methods achieves comparable accuracy with fewer nodes, lower CPU times, and smaller condition numbers.