A polynomial-reproducing Taylor transfer and Jameson–Schmidt–Turkel stabilization for the material point method

In the material point method, particle–grid velocity transfer schemes such as the fluid-implicit-particle (FLIP) and affine particle-in-cell (APIC) exhibit strong but opposite time-step dependencies: FLIP introduces less numerical dissipation but develops severe velocity oscillations as the time step decreases, whereas PIC-based schemes suppress oscillations but introduce increasing numerical dissipation under time-step refinement. This dissipation originates from the inability of PIC-type transfers to exactly reproduce the underlying velocity field, which leads to information loss during particle-to-grid projection. To overcome this limitation, we develop a lossless polynomial-reproducing velocity transfer of order k , which exactly reproduces any polynomial velocity field of degree ≤ k . The proposed transfer is derived through a novel recurrence relation that rigorously enforces the discrete recovery of derivatives up to order k and naturally generalizes the original first-order Taylor particle-in-cell (TPIC) transfer, resulting in a unified TPIC( k ) formulation with B-spline basis functions. By improving transfer accuracy, the proposed TPIC( k ) substantially reduces the dissipation inherent in PIC-type methods. However, the reduced numerical dissipation also makes the scheme more susceptible to spurious high-frequency velocity oscillations. To suppress these oscillations, we incorporate Jameson–Schmidt–Turkel (JST) stabilization into the proposed framework. The resulting JST-stabilized TPIC( k ) scheme achieves high accuracy, improved stability, and favorable energy preservation, as demonstrated in large elastic and elastoplastic simulations.