Локально-одномерная сплайн-схема четвертого порядка для квазилинейного диффузионно-кинетического уравнения с подвижной границей

Locally One-Dimensional Fourth-Order Spline Scheme for a Quasilinear Diffusion-Kinetic Equation in Polar Coordinates with a Moving Boundary

We present an extension of our earlier compact spline finite-difference scheme to the case of polar coordinates for a two-dimensional quasilinear parabolic diffusion-kinetic equation with a time- and angle-dependent moving boundary 𝑠(', 𝑡) = 𝑟0+(', 𝑡). The formulation uses a locally one-dimensional (LOD) construction and a coordinate mapping  = (𝑟−𝑟0)/(', 𝑡) to a fixed reference domain, which preserves conservation and consistently accounts for boundary motion. The discrete stencil is compact in the radial direction: control volumes have width 2ℎ𝑟. The scheme is constructed in a locally one–dimensional manner by applying one-dimensional cubic splines successively along the angular and then the radial coordinate. The overall spatial accuracy of the scheme is fourth order in both coordinates, the truncation error is 𝒪(ℎ4 𝑟 +ℎ4 '). The boundary flux at 𝑟 = 𝑠(', 𝑡) is incorporated in a conservative form and is compatible with mixed 2nd and 3rd-type boundary laws. Computational experiments confirm these properties. For manufactured (analytic) test problems we observe the design fourth-order convergence in both 𝐿2 norms. For problems without a closed-form solution we estimate the order by Runge’s refinement rule; the measured Runge indices satisfy 𝑝(𝑛) 𝑖,𝑗 (ℎ) ≈ 4 across grid refinements, which empirically verifies fourth-order spatial accuracy. The method remains robust on nonuniform meshes with ℎ𝑟 ̸= ℎ' and for time- and angle-dependent boundary deformations, while keeping a small stencil (radial width 2ℎ𝑟) and low memory footprint. These features make the scheme suitable for diffusion-kinetic applications where the geometry evolves due to surface-loss mechanisms.