An efficient numerical algorithm for simulating Swift–Hohenberg pattern formation on evolving geometric surfaces

In this study, we present a numerical method for solving the Swift–Hohenberg (SH) equation on evolving complex surfaces. The numerical method constructs the discrete Laplace–Beltrami operator directly on triangulated surfaces and their dual structures, and this removes the need for coordinate parameterization while it keeps geometric consistency. To handle time-dependent geometries, the operator is updated as the surface evolves. A stabilized splitting method is used with the Crank–Nicolson (CN) scheme, and the nonlinear term is solved in a semi-explicit way, which leads to a linear system that can be solved efficiently. Convergence tests confirm second-order accuracy in both space and time. Additional numerical tests show that the method remains stable under mesh distortion and different time-step sizes. Numerical experiments on various evolving geometries, such as spherical surfaces, expanding tori, surfaces with boundaries, anisotropically deforming surfaces, and complex shapes, show that the method captures pattern changes and describes the interaction between surface evolution and Swift–Hohenberg dynamics.