Shaoqin Zheng | Searcharxiv

arXiv2026

Decoupled energy-stable Runge-Kutta schemes of arbitrary order for the anisotropic phase-field dendritic crystal growth model

In this paper, we construct an arbitrary-order scheme for the anisotropic phase-field dendritic crystal growth model by introducing a time dependent auxiliary variable. By employing an algebraically stable Runge-Kutta method, the proposed scheme satisfies an unconditional discrete energy dissipation law. To reduce the computational cost, a matrix diagonalization technique is applied to the coupled elliptic system at each time step. This transforms the original system into independent elliptic equations with constant coefficients, which can be solved separately or in parallel. After the decoupling, the auxiliary variable is obtained from a uniquely solvable $q\times q$ algebraic system. For a fixed Fourier-Galerkin space, we further prove $q$th-order convergence in time for the scheme based on a $q$-stage Runge-Kutta method. Numerical experiments in two and three dimensions confirm the theoretical convergence rates and the discrete energy dissipation, and demonstrate the computational efficiency of the decoupled schemes. The effects of anisotropy, latent heat, orientation angle, and initial nuclei on the dendritic morphology are also investigated numerically.

math.NA↗

arXiv2024

High order conservative LDG-IMEX methods for the degenerate nonlinear non-equilibrium radiation diffusion problems

In this paper, we develop a class of high-order conservative methods for simulating non-equilibrium radiation diffusion problems. Numerically, this system poses significant challenges due to strong nonlinearity within the stiff source terms and the degeneracy of nonlinear diffusion terms. Explicit methods require impractically small time steps, while implicit methods, which offer stability, come with the challenge to guarantee the convergence of nonlinear iterative solvers. To overcome these challenges, we propose a predictor-corrector approach and design proper implicit-explicit time discretizations. In the predictor step, the system is reformulated into a nonconservative form and linear diffusion terms are introduced as a penalization to mitigate strong nonlinearities. We then employ a Picard iteration to secure convergence in handling the nonlinear aspects. The corrector step guarantees the conservation of total energy, which is vital for accurately simulating the speeds of propagating sharp fronts in this system. For spatial approximations, we utilize local discontinuous Galerkin finite element methods, coupled with positive-preserving and TVB limiters. We validate the orders of accuracy, conservation properties, and suitability of using large time steps for our proposed methods, through numerical experiments conducted on one- and two-dimensional spatial problems. In both homogeneous and heterogeneous non-equilibrium radiation diffusion problems, we attain a time stability condition comparable to that of a fully implicit time discretization. Such an approach is also applicable to many other reaction-diffusion systems.

math.NA↗