Abstract In this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.
An adaptive algorithm for steady convection-diffusion problems thatcombines a posteriori error estimation with conforming centroidalVoronoi Delaunay triangulations (CfCVDTs) is proposed and tested intwo dimensional domains. Different from most current adaptivemethods, this algorithm realizes mesh refinement and coarseningimplicitly at each level by nodes insertion and redistribution.Especially, the nodes redistribution is implemented through thegeneration of CfCVDTs with some density function derived from the aposteriori error estimators for the problem. Numerical experimentsshow that the convergence rates achieved are almost the bestobtainable using the linear finite volume discretizations and theresulting meshes always maintain high quality.
In this paper, we propose and test a novel diagonal sweeping domain decomposition method (DDM) with source transfer for solving the high-frequency Helmholtz equation in $\mathbb{R}^n$. In the method the computational domain is partitioned into overlapping checkerboard subdomains for source transfer with the perfectly matched layer (PML) technique, then a set of diagonal sweeps over the subdomains are specially designed to solve the system efficiently. The method improves the additive overlapping DDM (W. Leng and L. Ju, 2019) and the L-sweeps method (M. Taus, et al., 2019) by employing a more efficient subdomain solving order. We show that the method achieves the exact solution of the global PML problem with $2^n$ sweeps in the constant medium case. Although the sweeping usually implies sequential subdomain solves, the number of sequential steps required for each sweep in the method is only proportional to the $n$-th root of the number of subdomains when the domain decomposition is quasi-uniform with respect to all directions, thus it is very suitable for parallel computing of the Helmholtz problem with multiple right-hand sides through the pipeline processing. Extensive numerical experiments in two and three dimensions are presented to demonstrate the effectiveness and efficiency of the proposed method.
The convective Allen-Cahn equation has been widely used to simulate multi-phase flows in many phase-field models. As a generalized form of the classic Allen-Cahn equation, the convective Allen-Cahn equation still preserves the maximum bound principle (MBP) in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions preserves for all time a uniform pointwise bound in absolute value. In this paper, we develop efficient first- and second-order exponential time differencing (ETD) schemes combined with the linear stabilizing technique to preserve the MBP unconditionally in the discrete setting. The space discretization is done using the upwind difference scheme for the convective term and the central difference scheme for the diffusion term, and both the mobility and nonlinear terms are approximated through the linear convex interpolation. The unconditional preservation of the MBP of the proposed schemes is proven, and their convergence analysis is presented. Various numerical experiments in two and three dimensions are also carried out to verify the theoretical results.
We study in this paper a finite volume approximation of linear convection-diffusion equations defined on a sphere using the spherical Voronoi meshes, in particular the spherical centroidal Voronoi meshes. The high quality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computational experiments. In particular, we show that the L2 error of the approximate solution is of quadratic order when the underlying mesh is given by a spherical centroidal Voronoi mesh. We also demonstrate numerically the high accuracy and the superconvergence of the approximate solutions.
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