Abstract This article proposes and analyses a second-order consistent splitting scheme for the incompressible magnetohydrodynamic (MHD) system. The MHD equations form a strongly coupled multi-physics system that poses challenges for numerical simulations. Although considerable efforts have been devoted to developing decoupled methods for MHD system, achieving full second-order numerical accuracy remains difficult, especially in the treatment of decoupled magnetic induction, velocity and pressure fields. To address this limitation we propose a fully decoupled, linear, unconditionally stable second-order consistent splitting scheme. Implemented within the generalized scalar auxiliary variable framework the scheme is fully explicit while ensuring unconditional stability and uniform boundedness of the numerical solution. This stability facilitates a rigorous local optimal error analysis for all variables in the three-dimensional MHD equations. Finally, a series of three-dimensional numerical experiments confirm the full second-order convergence of the proposed scheme and illustrate the good performance of this method.
Abstract We introduce a new derivation of a barycentric sinc interpolant and prove its convergence for a class of functions that are analytic in a strip containing the real axis and nodes obtained from conformally shifting equispaced points. Such shifts are useful for the approximation of functions with locally steeper gradients. Corresponding conformal maps are given and experimented with, also for functions with several steeper gradients. The convergence is compared with that of a linear rational interpolant at shifted Chebyshev points. Formulas for the derivatives are also given and tested.
Abstract We construct and study projection-free Gauss methods for harmonic maps into spheres, extending the Crank–Nicolson scheme to arbitrarily high order. Existing projection-free methods are limited to second-order accuracy; in contrast, our schemes attain high-order temporal accuracy while preserving the unit-length constraint and ensuring discrete energy dissipation unconditionally. We furthermore introduce practical linearized variants of the base schemes via fixed-point iteration, which preserve the constraint up to machine precision when sufficiently iterated, and achieve higher-order accuracy in the constraint violation even with a fixed number of iterations. Numerical experiments demonstrate that the proposed methods maintain high temporal accuracy and effectively control the constraint violation; notably, preserving the unit-length constraint up to machine precision enables highly accurate steady-state computations with large time steps, highlighting the robustness and efficiency of the methods.
Abstract We investigate the convergence properties of the sweeping preconditioner with sequentially semiseparable (SSS) Schur complement approximation proposed by Qiu et al. (2015b, Efficient preconditioners for PDE-constrained optimization problems with a multilevel sequentially semiseparable matrix structure. Electron. Trans. Numer. Anal., 44, 367–400). The sweeping preconditioner is obtained by replacing the Schur complements in the block LU factorization of the global linear system by SSS matrix approximations with low off-diagonal rank. We show that the spectrum of the preconditioned system is confined within a circle centred at $(1, 0)$ and derive an analytic estimate of the radius of this circle. By appropriately tuning a parameter in the sweeping SSS preconditioner the radius can be made arbitrarily small. Our analysis results hold for a broad class of linear systems arising from discretizations of two-dimensional partial differential equations with uniform structured grids, irrespective of whether the system matrix is symmetric or unsymmetric, definite or indefinite and even with saddle-point structure. We demonstrate the efficacy of our analysis by numerical experiments.
Abstract The internality of quadrature rules ensures the approximation of integrals whose integrands are defined only on the support interval of corresponding measure. Previous studies addressed the internality of averaged Gauss and optimal averaged Gauss quadratures for Jacobi measures modified by a linear factor or divisor. In this paper we extend the analysis to Jacobi measures modified by a quadratic factor or divisor of the form $(t - z)(t - \overline{z})$, where $z = x + iy$ with $y> 0$. We prove that, under certain conditions and for sufficiently many nodes, these quadratures are internal. Special focus is given to Chebyshev measures of the first and third kinds.
