Rakhmanov A.Yu., Rusakov S.V. Locally One-Dimensional Fourth-Order Spline Scheme for a Quasilinear Diffusion-Kinetic Equation in Polar Coordinates with a Moving Boundary
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https://doi.org/10.15688/mpcm.jvolsu.2026.1.6
Alexey Y. Rakhmanov
Postgraduate Student, Institute of Physics and Mathematics, Perm State University
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https://orcid.org/0009-0001-3332-2124
Bukireva St, 15, 614068 Perm, Russian Federation
Sergey V. Rusakov
Doctor of Sciences (Physics and Mathematics), Professor, Center of Applied Mathematics and Physics, Perm State University
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https://orcid.org/0000-0001-6862-1100
Bukireva St, 15, 614068 Perm, Russian Federation
Abstract. 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(ϕ, t) = r0+δ(ϕ, t). The formulation uses a locally one-dimensional (LOD) construction and a coordinate mapping ξ = (r−r0)/δ(ϕ, 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(g4r+h4ϕ). The boundary flux at r = s(ϕ, 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≠ hϕ 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.
Key words: Laplace operator, image contour, contour region graph, image processing, feature extraction.
Rakhmanov A.Yu., Rusakov S.V. Locally One-Dimensional Fourth-Order Spline Scheme for a Quasilinear Diffusion-Kinetic Equation in Polar Coordinates with a Moving Boundary is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Mathematical Physics and Computer Simulation. Vol. 29 No. 1 2026, pp. 66-76
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