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In this paper, a novel numerical algorithm for efficient modeling of three-dimensional shape transformation governed by the modified Allen-Cahn (A-C) equation is developed, which has important significance for computer science and graphics technology. The new idea of the proposed method is as follows. Firstly, the operator splitting method is used to decompose the three-dimensional problem into a series of one-dimensional subproblems that can be solved in parallel in the same direction. Secondly, a temporal p-adaptive strategy, which is based on the extrapolation technique, is proposed to improve the convergence order in time and preserve the computational efficiency simultaneously. Finally, a parallel least distance modification technique is developed to force the discrete maximum bound principle. The proposed method achieves high precision and high efficiency at the same time. Numerical examples include the effectiveness of the p-adaptive method and the bound preserving least distance modification, and a series of complex three-dimensional shape transformation modelings. • The operator splitting method is used to solve 3D shape transformation PDE. • A temporal p-adaptive strategy is developed to improve computational efficiency. • A least-distance modification is developed to force the discrete maximum bound. [ABSTRACT FROM AUTHOR]

In this paper, we present two structure-preserving-doubling like algorithms for obtaining the positive definite solution of the nonlinear matrix equation X + A H X ¯ − 1 A = Q , where X ∈ C n × n is an unknown matrix and Q ∈ C n × n is a Hermitian positive definite matrix. We prove that the sequences generated by the algorithms converge to the positive definite solution of the considered matrix equation R-quadratically. In addition, we also present some numerical results to illustrate the behavior of the considered algorithm. [ABSTRACT FROM AUTHOR]