Explicit stability of nonlocal operators for transient heat conduction: the role of the constant basis term

Purpose Nonlocal strong-form operators provide a meshfree spatial discretization of transient heat conduction, avoiding mesh generation on geometrically complex domains. Under explicit time marching, stability is strongly influenced by the local moment matrix used to reconstruct the spatial derivatives. The nonlocal operator method (NOM) and the peridynamic differential operator (PDDO) assemble their moment matrices from the same weighted data but differ in their treatment of the constant basis term, whose implications for explicit time stepping have not been thoroughly studied. This investigation pinpoints that influence through an explicit Euler–NOM scheme (EE–NOM). Design/methodology/approach Boundary ghost nodes preserve stencil symmetry on uniform grids, where stability is assessed by von Neumann analysis, whereas irregular and graded sets are assessed through eigenvalue and pseudospectral diagnostics. The scheme is tested alongside the explicit Euler–PDDO (EE–PDDO) on perturbed, nonconvex, curved-boundary, cracked and nonsmooth configurations, with hardware-independent operation counts. Findings Both schemes attain second-order convergence on smooth benchmarks, where the NOM yields lower errors, while both show first-order convergence with indistinguishable errors on the regularity-limited benchmark. Critical Fourier numbers of both schemes exceed the finite-difference and consistent-mass finite-element limits. Spectral admissibility holds for both operators on generator-quality graded clouds, but only the NOM remains admissible over a wider range on anisotropic axis-aligned lattices; no significant growth is observed on jittered clouds. Originality/value In the two quadratic reconstructions compared here, the treatment of the constant basis term markedly affects conditioning, non-normality and spectral robustness. On irregular and graded node sets, the constant-free difference form makes the NOM the more robust operator for explicit transient solution.